There's no such thing as a "proper" mobile game but if there was, Final Fantasy III would't be it.
For a game to be a "proper" mobile game, it must have been made specifically for mobile. Examples: Clash Of Clans, Puzzles and Dragons, Candy Crush...etc.

If you like shooters, the Modern Combat series by Gameloft is very good.

I thought you'd meant you'd made one and got excited that you might have some tips...

The talents of those developers are being wasted by mobile.

From my experience, only good use for mobile as gaming platform is to install emulators and play old JRPGs I won't bother with on big screen.

i actually agree. there are plenty of good mobile HD games, the only problem is the controls. touch control still cannot replace buttons, at least for me

If only Square Enix knew how to price their games, I'd get into mobile gaming... Not paying 15€ for an inferior experience, when I can get the better experience for 5-10€ depending on the moment. (And considering I already own it/them.) Sure, on the go would be nice but it's still too much for me.

Also got an Android, but I mainly mobile-gamed on my iPod Touch (don't do it that often anymore).
Even though these are for iOS, I recommend you visit: toucharcade and appspy.

What games do I play on my Xiaomi?
WarLight
PuzzleBits
Age of Civilizations
Super Monsters Ate My Condo!
Smash Hit
Can Knockdown 3
Voxel Rush
Into the Dead
Star Acres
Crossy Roads
Minecraft PE (this Xmas with my nephew, haven't touched it since)

And recently I've been trying thomas was alone, even though I've already played it on PC.
Note sure I will like it, I prefer arcady touch games for my mobile needs.