I’ve been punting on this for a while but it came up as neatly tangential last week, so here we are. Let’s talk about mobile/tablet apps.
I’ll queue up three categories:
The core questions are what do you play, which do you most respect, and what would you recommend if someone was coming in blind and wanted a starter library?
I’m not really much of a computer game or mobile game player, but I do have a copy of an implementation of Simon Tathan’s puzzle collection (Simon Tatham's Puzzles | F-Droid - Free and Open Source Android App Repository) for mental gear-changing. I don’t really do any board-gamey mobile things except for dice rollers, first player randomisers, and score keepers (https://play.google.com/store/apps/details?id=com.sbgapps.scoreit is the best I’ve found).
I have 3 devices on which I play “apps” in the widest sense. Over the years I bought a whole bunch of apps to play. First which ones am I playing these days:
I have reduced my on-the-phone playing to almost nothing because most boardgames don’t play that well on the small screen. I still own a bunch of boardgame apps on Android because I used to have an Android tablet but I switched a few years ago.
Also obviously BGStats 
Before BGA, I would buy almost every boardgame app that came out especially because they used to be in the 5€ range and not like 15-20€ as is now quite common.
Ones that I have either played a lot at some point or still play now are:
There are probably more but… iOS sometimes uninstalls apps on its own.
Also on my tablet at all times:
What I don’t play anymore are various mobile games that require some kind of internet and that allow ingame payments. I once wasted some good money on some EA Star Wars money sink and I regret that. Also no longer on my devices for the same reasons: Hearthstone.
I have also uninstalled Fallout Shelter because it gets too addictive and I needed a break. Games that make me click every few hours for something are also generally gone: “your quest ends in 10 hours come back then” → I have a Steam Account filled with games that don’t make me wait.
I will try to stick to what I consider boardgame adjacent and my favorite boardgame implementations because my game library is huge.
And some actual boardgames worth mentioning:
As for starter library… that is really hard. Why would someone need a starter library of boardgames? But if I had to uninstall everything. Here are the ones I would definitely install first:
I think all the good board game mobile apps have been covered, so I’ll just add Slice & Dice, Hoplite, and Pixel Dungeon (variants) as good sort-of boardgame-adjacent games.
My apps are very unplanned and certainly not the best out there, but
Better as an app:
Sagrada (shows you the valid spaces to go on)
Splendor
Elder Sign: Omens (although the difficulty is set to impossible and real dice would never fail you this often)
Onirim (definitely would not have learned the strategy fast enough if I had to do all that shuffling by hand)
Good enough:
Lanterns: The Harvest Festival
Lotus
Prune
Tokaido
Indian Summer (cute music and squirrels)
Talisman and many expansions (better on Steam than phone)
Apps that turned me off the game:
One Deck Dungeon (you don’t need to endlessly shuffle cards and can get down to the actual game quicker. Unfortunately, as @yashima said, that just shows you that the game isn’t very good, and makes it obvious that you need to lose 20 times to get enough xp to be close to statistically winning)
Galaxy Trucker (…no)
Lords of Waterdeep (okay. That’s… fine, then?) (on Steam)
Takenoko (same)
I learned Backgammon on “Backgammon Mighty”, and The Royal Game of Ur on “The Royal Game of Ur”, although there are many versions of that out there and only a few of them look good visually.
Shout out to “Ultimate Jewel: Tutankhamun” for being the random jewel game I put on my phone years and years ago and have never stopped playing.
I enjoy most board game apps I have played, though some are better made than others. I like them because even if they aren’t the best, there is usually an AI that I can play against, which means I can play the game whenever I want!
Some of the better apps I like are A Game of Thrones, Castles of Mad King Ludwig, Istanbul, Tokaido, Onitama, and Root. Honestly, I think the Root app is a great way to learn the game as the tutorial is pretty good.
I have not played enough of the Concordia app to have enough of an opinion on it yet. Isle of Skye is good.
Apps I also enjoy but are not quite as well done are Unmatched (naturally) and Lords of Waterdeep. Unmatched has horrible AI and still has some weird bugs, as such it is unlikely it will ever get more fighters for it. Lords is just a little hard to follow on a small screen, but works well otherwise.
Twilight Struggle is great, but has made it very clear to me I want nothing to do with the physical game. Way too much to maintain and I still struggle with calculating what I need to roll for coups and such.
Similarly, Elder Sign has killed any interest in playing the physical game. Though I have to agree with @SteveB_uk that the die rolls seem horribly weighed against you in the app.
Galaxy Trucker is fine, but gets repetitive. At least it’s humorous.
For time wasting, I will often play an app called Einstein’s Riddle Logic Puzzle, which is just a logic game with a grid and a bunch of hints to let you determine the correct solution. I play on hard difficulty and usually finish a puzzle within 10 minutes, but have the occasional screw-up when I need 30 or so.
my Einstein riddle has insane difficulty mode with 9x9 and larger grids. i have spent as much as 4 hours on some of those puzzles. and sometimes they take just 15 min. it’s very soothing to do these when something is stressing me. the shorter ones with smaller grid are good for waiting rooms as well
I would like to note that I really enjoyed the games in my “addictive and then not category” but sometimes I just overplay something by a lot and then suddenly the fun is gone.
I have Oceans on my phone, which I can play a lot at times, now I am on a “rest” period from it. Other than that I play on Steam.
My list on Steam (from memory):
-Gloomhaven: the last couple of months or so my mate Ben and I discovered Jaws of the Lion was on sale and we bought it, having a weekly online session of one or two scenarios every Tuesday evening, we have played it through. Definitely beats playing it solo with more than one character, it can be a bit too much managing two or three characters (it’s doable, but way less efficient). We finished the main missions last week, so it is likely that we will continue on tonight with normal Gloomhaven.
Other Steam games:
-Root
-Lords of Waterdeep
-Through the Ages
-Raiders of the North Sea
-Wingspan
-Pandemic
Those first four I play very often. The other, no so much.
Wingspan less as of late. Through the Ages being my latest addition to the list in the last summer sale.
I also have Spirit Island (sorry, to me the app looks so ugly somehow) Terraforming Mars (I have fallen asleep playing it, sooo slow) Blood Rage (recently acquired on the same sale, still haven’t had a go at it with all this avalanche of Olympic sports the last two weeks) Splendor (got tired of it, but it has interesting challenges)
I’m definitely interested in the Due Imperium one, but I am waiting for the next price drop chance.
I have recently uninstalled all of the games from my phone to remove some of the temptation to waste time when I should be doing other things 
I can also recommend Appblock and Blocksite for anyone who has similar issues.
Having said that :
More likely to play than the physical game
The first two because of the serious reduction in faff provided by playing the digital version. The others are just games that I prefer to play solo and I prefer digital implementations for that.
For playing while waiting for the train (or similar)
I like these in person as well.
To help learn the rules to the ‘real’ game
We made so many rules mistakes the first few times we played Scythe 
Puzzles (Not board games)
These are basically logic grid puzzles where you end up with a picture if you fill in the grid correctly (also known as picross, I think)
Apps that I should never install on my phone again
I’m sure these games are designed by a psychologist to make them as addictive as possible
I’ll try to spare the commentary.
The best puzzles I’ve encountered on IOS:
Preferred Time Wasters
(Note that these never survive forever but each had a good run)
I feel like I’ve been kicking around “best puzzle” forums and ios sales longer than most, which surprises me around here. I’ve also got most every board game (apart from the newest ones).
Seems the tablet board game discussion has been light so I’ll sit on it like a full grown gorilla.
Board games that had their day or still stick:
Little Morsels
Session
A Real Sit Down
If that seems a bit unfiltered, it’s not. That’s maybe half of the games I have installed and doesn’t include anything I’ve deleted (like Sagrada) or lost (like Ghost Stories).
For the especially curious, here is everything I’ve played that’s been deprecated:
Just haven’t tried yet:
Yes, assuming you mean paying for the game instead of just playing the free demo. There aren’t any further paid unlocks afaik (and I love that you can simply unlock all the progress content from the menu).
Mostly just Straight Cribbage or Libre Sudoku.
On PC, the Hexcells series is easily one of the best puzzles I ever played.
I used to to this many many years ago when games on mobile were more scarce and I played more on my devices and less on my table. Back then my tablet was an Android and Google was not yet so very clearly stepping away from the „Don‘t be evil“ motto.
These days I am more inclined to buy a (board-)game or even puzzle app on Steam because I anticipate that my laptop (which has already replaced the PC/gaming machine) and tablet will merge in one of the next hardware generations a few years from now.
Also I tend to play even puzzles TOO MUCH and I am trying not to do that anymore. Things I don‘t mind doing TOO MUCH are reading and playing games on my physical table. Not that I am actively trying to limit my screentime but I have experienced the addictive nature of many games too often … see what @Whistle_Pig wrote about Candy Crush. I played that one, too.
During our vacation with friends in May, most of the family more or less addictively played some kind of Sonic mobile game. I didn‘t really mind them doing it but I also didn‘t want to have that kind of app on my phone anymore. And so I stopped trawling the platforms for new experiences because the good ones are so rare and the numbers of the „traps“ are so great.
Zachtronics are always these logical programmatic games. SpaceChem is fun too. Opus Magnum is super pretty though
Zachtronics games are great. But I always seem to burn out on them, or get stuck, and if I try to return to them I find I’ve forgotten too much of the knowledge that got me to the point where I got stuck, and I’ll be more lost than when I left off. Still amazing just for that first period of addiction though.
(And yeah, Magnus Opum might be the only one I ever “finished”, I think because progressing through the story was deliberately made easy.)
Oh yes, I had this and the sequel!
I’d recommend 80 days as a text adventure (of “Around the world in…”) but if you actually want to make it to your destination on time there’s only one or two routes you have to rush down.
Curses, sometime in the last 6 months they upped it from $7 to $9. And there doesn’t appear to be a cross platform unlock. (ios tablet and android phone, here). Oh well. I have a lot of google play credits stored up and this has been recommended frequently enough that it must be go time.
Can’t recommend Little Inferno enough here… it’s the same designer as World of Goo and Human Resource Machine. It’s a gamified morality tale with a easy to agree with takeaway that these games addict you while eroding your soul, and we never see what it is actually costing us…
We all know that, a bit. But Little Inferno transfers it in an Inception-like way that gets deep inside you. It’s been a good insulator against those types of experiences ever since.
The creator argues that games should have a beginning and an end, and I wholeheartely agree, even for puzzle games that can be procedurally generated every day.
These were the good old days when board games were required to go on sale to 0.99 once per year. Gotta get them all. I’m still angry at Temple Gate Games and RFTG - I think that is the first game that refused to ever go on sale and the market has shifted with it.
Of course, most of those amazing early developers (Sage, Big Daddy) have gone out of business, so maybe there’s something to avoiding 0.99 sales…
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Reading comments and looking at my stuff, other bg apps I didn’t get on with:
Onirim, Elder Sign, Tokaido, Ascension (I hate this one, sorry Yash), Shards of Infinity (the attempt to improve Ascension, but I’m in camp Star Realms), Ganz Schon Clever, Mystic Vale, Potion Explosion, Caylus (just not my game), Tikal, Brief History of the World
Games I only have on Android and are ok:
Takenoko, Red 7, Ingenious
Another game I haven’t tried: Metro
Another good puzzle/time waster: Hidden Folks / Hidden Through Time. It’s essentially a where’s waldo type world with stick figures and it was just delightful. It has a beginning and an end.
Can’t believe I’ve owned and played all these over 12-14 years?
Oh yeah, the Tsuro app was actually a really good implementation if I remember right.