There are a fair few remakes and multiplats that have made their way to mobile and worth playing.

Final Fantasy IV, V and VI
Baldurs Gate 1 and 2
GTA III, Vice City and San Andreas
A Bard's Tale
Star Wars: Knights of the Old Republic
The Walking Dead
Xcom: Enemy Unknown

I'm actually really surprised by how well some of these controlled on a tablet. It was a struggle on a 5 inch phone for GTA, but it should be OK on a Note.

As for original, "proper" mobile games:

Deus Ex: The Fall
Ravensword: Shadowlands
Aralon
The Room 1 & 2
Asphalt 8

mobile games are the shit. i don't get the stigma, tbh. the two rayman games, rayman jungle run and rayman fiesta run, are almost as good as their console counterparts. hitman go, a board game puzzle version of hitman, is terrific. monument valley, an ipad game with colorful graphics and captain-toad-like geometric stage design is equally awesome. device 6 is one of the best games i've played recently period, and anyone who enjoys text adventures will appreciate it.

people keep on talking about mobile games as tho they're the end of the world. but the fact is mobile games are far more diverse than candy crush and angry birds (which are incredibly shallow and designed to rip you off). the best developers (simogo, for example) are fully committed to the best principles of good game design, or as committed as anyone else developing for consoles.

A handful of mobile games are good. Anyone who hasn't yet played the first Plants vs Zombies should get the HD iPad version. A lot of quality there.

Try to avoid a lot of the games in top seller lists on iOS and Android. That's where most of the garbage is found.

Chaos Rings seems to have been recommended already. Another I'd suggest looking into is Terra Battle. It's Mistwalker's free to play mobile title, and it's actually pretty well put together. I've enjoyed my experience with it so far.

Well, I wouldn't call it "proper" mobile game , since it's a console port, but welcome to mobile gaming nevertheless.

Here are few suggestions, if you like puzzle games:

Room 1/2
Monument Valley
Bad Piggies
Cut the Rope
God of Light
Where's my Mickey/Where's my water
Osmos
Threes
Puzzle Retreat
Tiny Thief
Back to Bed
Crazy Machines

and few more if you like local coop (tablet recommended):

Slingshot Racing
Draw Race 2
Hey That's my Fish
Through the Desert
King of Opera
Starlink

Of course, there are many more in other genres, but this is what I play the most on mobile.

If you don't mind ports or remakes then I can recommend quite alot of games.
Paid
The world ends with you - ios8 is still not supported
Phoenix wright trilogy
Phoenix wright dual destiny - looks soo much better than the 3ds version.
Knight of the old republic - it is basically the xbox version. No downgrade in graphics or fps.
Dragon quest 5 - just got released yesterday. So far, the user reviews are quite positive but it looks like there's no landscape mode... Again.
Oddworld strangers wrath - its on sale $3.99. But imo controller is a must in this game.
Monster hunter freedom unite hd - surprising, the controls are fine. You can lock the camera to your object monsters.
The banner saga - awesome game!! Imo a must buy
Minecraft PE
Combo crew - great beat em up. chun li, viewtiful joe, ryu, ken and blanka are sold seperately though.
Kingdom rush origins - my favorite td.
Dragon quest 8 - they inexplicably lock the fps to 20, no voice acting, no landscape mode. Its still decent but i suggest you watch some youtube gameplay first and see if you can stomach the frame rate.
Telltale games are also available but i haven't tried the ios versions. 'M thinking of buying the tales of the border lands after they release all the episodes.

Free
Spiderman unlimited - endless runner. Great artstyle. IAP isnt that bad.
Injustice - great graphics. Constant grinding is needed to unlock character.
Asphalt 8

I don't think my low standards were ever in question seeing how I played through all 3 games of FFXIII and even liked a bit of them.

Also, I own a Wii U. So there's that.

I really liked Flappy Bird, Temple run, nin jump. extreme landings and a couple of other ones which were also really good.

The only mobile game I really enjoy is Fruit Ninja! I just find it way too damn addictive!

I beaten final fantasy tactics on my iPad. Very good game. The only part that frustrated me was the wiegaf/ demon/ assassin's fight. Too ridiculous. After that the game was easy.