The latest version of the iPhone OS allows you to have 11 pages of 16 icons, plus the four permanent icons at the bottom of the screen, for a total of 180 apps. You can actually have more than 180, though only 180 icons will be visible.
To get to the invisible apps, just do a search (press Home from the home screen), and type part of the non-visible app's name. I don't know how many of these invisible apps you can have, as I have not tried more than three so far.
[robg adds: As far as I know, the limit is simply based on the available space on your device. Long before that point, though, you may run into a limit with your ability to remember the names of all the invisible apps.]
Mac OS X Hints
http://hints.macworld.com/article.php?story=20091026034353289