How can I get more saved-game slots in Portal 2?

I don't think you can do what you ask, but unless you want to keep saves to access extremely specific parts of levels, you really don't need any more saves. Once you've finished a level you can always go back to it afterwards and play it again. So if there are some puzzles you like, you can skip straight to them (or to the beginning of the chapter, anyway).

Addendum from comments: Remember, you can always quicksave (F5) and quickload (F9), which is much faster. But you always overwrite your previous quicksave, so be aware of that. However, due to the design of the game it's essentially impossible to lose any progress short of a complete wipe of your saves.

You can save over previous save slots - if you find you run out just keep some useful ones.

Remember you shouldn't need very many - only enough to complete a single level, then maybe again to get the achievements you missed first time around.