There never has been a ss3 gogeta.
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No. Super Saiyan 5. That level was never attained. And since the series is over, it never will be. It pretty much doesn't exist.
Super Saiyan, Ultra Super Saiyan, Ultimate Super Saiyan, Full Power Super Saiyan, Super Saiyan 2, Mastered Super Saiyan 2, Super Saiyan 3 and 100% Super Saiyan 3. In GT, Super Saiyan 4 exists. Some belive that SSJ4 Goku fused with Shenron and absorbing the dragon balls is the equivilant of "SSJ5" which only exists in DB AE/AF which are fanmade creations.
Episode 230 "Super Saiyan 3"
movie 8 the legendary super saiyan
Goku does not turn super saiyan 6 or 5 in fact. and it only goes up to super saiyan 3 in the manga and super saiyan 4 in the cartoon.