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 $US77 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!]