Fat, and I want it to play burned games. Mainly PS1 games since I lack a DVD burner.
It depends on the method you're choosing and what you're willing to spend.
If you chip it, you're right off and no other mess. If you know how to chip, that is.
My method, while a bit of a process, was also nice and cheap.
[spoiler]The one I used involved a program known as Free MCBoot. I needed a copy of 007 Agent Under Fire. I then ripped the ISO, patched it, played the original, used a make shift swap tool to remove the game just before the second mission, insert my patched DVD, and launch ulaunchELF. From there you install Free MCBoot to a memory card. If you're looking for PSX Backups, you use a program called PSXLauncher which will require you to start with any regular PS1 disc and then swap it for your backup.
Granted, this method also require you take the front of your disc tray off, which can cause you to actually break the front plate if you're not careful. It's a bit of a process, but it's paid off wonderfully for me and if I ever need a replacement, I'll be able to just transfer my memory card right over to the new fat or even slim.[/spoiler]