I've never thought of myself as a gamer, but for each phone I've had I've always found one or two games I love and play obsessively.
On the Nokia 3650, the games were Tris and Interstellar Flames.
Tris is the original Tetrus, and it was free (because, I seem to recall reading, it was developed in the Soviet Union). (A few years after its introduction, Tris became unavailable in the west because Electronic Arts claimed the copyright, but now, apparently, it's available again - to those few people who still have Symbian Series 60 or Palm devices.) It was simple and awesome and addictive.
Interstellar Flames was the first 3D game for mobile that I know of. It was developed by Xen Games for Symbian Series 60 devices, such as the 3650, and was pretty compelling in 2004. It was also free, because back then games were easily hacked and the developer presumably didn't see any way to prevent that, so they just gave it away. Here is a link to Xen Games' original page for Interstellar Flames:
In 2004-5 a 3D First Person Shooter became available for later Symbian devices. Called Daedalus 3D, it promised a level of depth and realism that had never before been achieved.
My Nokia 3650 was outdated by then, but I wanted to try the game. This phone used part of its limited RAM for storage, and consequently I could install the the game but not run it. I wrote to the developer in Italy, and not only did he reply, he created a multi-part install file for my phone and sent it to me along with special instructions. I actually got the thing running in a limited fashion. It was pretty cool at the time.
For my next two phones, the Nokia 6620 and Nokia N95, the game of choice was Raging Thunder, a 3D car racing game from Polarbit.
Unlike previous racing games, this one actually had somewhat realistic physics and road feel, and was pretty addictive.
The favored game on my next phone, the iPhone 3Gs, was a port of my favorite from the arcade days, Atari Centipede. I had many hours into Centipede before the fun finally ran out.
Since January I've been playing the Infinity Blade games (I & II) on iPhone 4s. I actually think the first version of the game is superior to the sequel, but the sequel offers more depth and more extended play.
I've pretty much got Infinity Blade II beaten, and am looking for something of similar quality to replace it. Any suggestions?
What are you playing on your phone or mobile device?