Ever since my wife recently got us iPhones, I've been using it more and more to play games. So far I've downloaded the following:
 
- Tetris
- Call of Atlantis (Match 3 Puzzle)
- Around the World in 80 Days (Match 3 Puzzle)
- Rebirth of Fortune (Tactics)
- Transformers G1 Awakening (Tactics)
- Secret of Monkey Island SE
 
The puzzle games are great, the tactical games are good, but a bit awkward, and Monkey Island isn't too bad in this format & still a lot of fun to play. So, can any of you out there reading this recommend any other great iPhone games & don't say "Angry Birds".

The War of Eustrath -  Mech version of Fire Emblem  
Tomb Treasure - match type puzzle game 
 
 
Eustrath is one of the best games I have played on a portable in years.

Papa Sangre

I really liked Infinity Blade if you want to see the Unreal engine at work on the iOS. It's pretty much a medieval fantasy Punch-Out with RPG elements.

Battleheart

Game Dev Story

Shibuya

GeoDefense Swarm

Infinity Blade

Tetris is a classic. 
 
Super K.O. Boxing 2
 
Simple Physics 
 
Millionaire City 
 
Around The World in 80 Days

Tiny Wings.

Dungeon Raid.

GameDevStory.

Plants vs. Zombies.

Robot Unicorn Attack.

All these are games I own and can happily recommend to anyone, even without incurring Angry Birds or Fruit Ninja based wrath.

Great Little War Game 
 
The Settlers 
 
Civ Rev 
 
Trenches

1. Plants vs. Zombies
2. Catan
3. Civilization Revolution
4. Monopoly (or Monopoly: Here and Now)
5. Carcassonne
6. Battleship
7. Uno
8. Wheel of Fortune
9. Game Dev Story
10. Peggle
11. Doodle God
12. Pocket Tanks
13. Game of Life
14. Scrabble
15. Bejeweled 2
16. Chuzzle
17. Auditorium
18. Oregon Trail
19. The Creeps
20. Trainyard (also has a lengthy free version with puzzles not in the full game)
21. Hector: Badge of Carnage - Episode 1
22. Puzzle Agent
23. KenKen (from Capcom)
24. Cut the Rope
25. Fragger
26. Cat Physics
27. Phoenix Wright: Ace Attorney
28. Trenches
29. Dark Nebula
30. Lock 'n Roll
31. Traffic Rush
32. Gyrotate
33. Sky Burger
34. Shibuya
35. SET
36. Zonix
37. Yahtzee Adventures
38. Doodle Fit
39. Wolfenstein RPG
40. Doom RPG
41. Charmed
42. Geared (or Geared 2)
43. 4 in a row (basically Connect 4, but free)
44. Topsy Turvy
45. Vizati
46. Infect Them All
47. Puzzle Quest
48. Hot Springs Sim
49. The Heist
50. 7 Words

There are some to get you going. I underlined my favorites.