It's Not Really A PSP, But It'll Do

This "PS One Handheld Game Player" is in no way endorsed or supported by Sony. That doesn't stop it playing PS1 games, though.

For USD$77 you can buy the unit, based off an older PSP knock-off that was actually a portable TV and media player. This newer model, though, is more in line with the spirit of the design it's knocking-off, as it claims to be able to run PS1 games (along with what looks like a few basic multimedia features).

You'd obviously have to do so by uploading the games to the unit, which puts it in a legal grey area, but if you're the type to make back-ups of your actual PS1 discs then this might be worth looking into.

Handheld PSOne Released - PS One Handheld Game Player [DCEmu, thanks wraggy!]