What people had been speculating about a possible Apple console before the announcement of the iPhone have on one hand turned out to be false, but on the other they have been proven right in a way. Because although Apple didn't make a competitor to the Xbox 360, the Playstation 3 and the Wii, the iPhone turned out to be a strong competitor of the Sony PSP and the DSi. The company made a smartphone that's very much capable of being used as a handheld gaming console.
Initially only the independent developers saw potential in the iPhone, but as the first small teams or even individuals made their first million from their free games for ipod touch, the big publishers got really interested, and nowadays almost every larger title has a port for the iPhone, be it either the Force Unleashed or the Need for Speed: Shift. Of course these games are much simpler on the iPhone than on the bigger consoles, but the business is rolling.
Not to mention that the phone has a much stronger hardware than the first personal computers had, thus classics like Doom or Wolfenstein can be easily run on it, and since people still love these programs, there is an awful lot of money in these titles as well.
But people are not only seeking after the big names, they also love the smaller masterpieces that have just come out of the blue. Games from almost every genre can be found in the App Store, from tower-defense to platforming, there are hundreds of adventure and shooting titles, as well as numerous forms of Tetris and pinball. Just for comparison: the Sony PSP has around 600 games released, the Nintendo DS 4000, while on the iPhone there are more than 20000 games available, and that's a very incredible number to say the least.
Of course these stats need a little explanation, since this number includes games that were being developed for about 2 days like Paper Toss, in which you have to throw pieces of paper into the bin.
So all in all the iPhone is still very much away from the PSP and the DSi in terms of depts and abilities, but I'm sure everyone could easily find at least 10 titles from the 20000 that he could enjoy very much, not to mention that a large portion of these games are either free, or cheaper than 1 dollar, and these are pretty strong arguments in favor of the iPhone for a consumer interested in games.