Download the APK directly (I forget how exactly, but you can download it from Play) and try manually installing it. The version compatibility check for 13 is pseudo-whitelist based (basically checks if app has been compiled with one of the recent frameworks targeted), so most of those "version too new" errors come up for apps that simply weren't checked for compatibility. Many of them still work fine, or fine enough.
Also you can get it on other app stores which don't enforce version checking. The version I have comes from the Amazon App Store, for example. Then again my phone isn't running 13.
A lot of really good older games have been ported to mobile and work fantastically. Knights of the Old Republic, Neverwinter Nights, and Baldur's Gate just to name a few. Hell, you can play Baldur's Gate multiplayer, cross-platform with PC players even! There's also fo2.exe on the Play Store, which lets you play Fallout 1 & 2 on mobile provided you have access to the files from the PC versions. Oh, and Doom ports galore, of course.
As far as mobile-only or mobile-specific games, my favorite is probably Caves, a surprisingly well done and completely free roguelike.
Also an honorable mention goes to Organ Trail, a zombie-themed take on Oregon Trail that is, frankly, just a flatly better version of the game in almost all regards. The only reason it's not my favorite is because the controls on mobile feel awkward compared to the PC version. Still fun though.
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