No. The closest thing you can get to a 4 3/4ths grip is if you take a 5/8ths and add a leather replacement grip and 2 thick overgrips. That should get you something like a 3/4ths grip.
The thing here is that they go up every size until 1/2 and then they start skipping some after 5/8ths.. First, there's 1/8, than 1/4, and then 3/8, then 1/2, then 5/8ths and I think there's nothing larger until 5 on the spot. Could be wrong though.
I also noticed you're from CT. Whereabouts? I'm from Fairfield county.