Wondering if the PS3 has a way for game developers to make free / affordable games available to their consumer base? Is the only way to get your game on the PS3 to go through an approval process which may cost money?
No, there is nothing equivalent to XNA on the Xbox 360.
There is LittleBigPlanet though, depending on how complex your game needs to be ;)
No, there is no Creators Club equivalent for PS3, however Sony does have PS3/PSP Minis. There's a [rather large] outlay for a devkit though. More details can be found in the Minis Guidelines available on the License Enquiry Page
If you somehow have a PS3 that hasn't been updated in a while you could use OtherOS to run some stuff under Linux. The GPU is disabled though, so nothing needing a lot of horsepower unfortunately.