I am planing to rebuild my 350 this winter.  I was wondering what would be the max hp possible with the automatic 3 speed transmission that came with the 1976 c3.

It's not the horsepower, it's more about how much torque it can handle.

Since the Turbo-Hydramatic transmission withstood the torque of the big block Corvettes in the early '70s, I suspect it will take anything that your 350 will put out if it's naturally aspirated.  Of course this assumes that GM didn't do anything to weaken the transmission in the later 70s.

It is going to depend on your driving style. If you lean on Any drivetrain, eventually it will break. Misuse it, Lose it!  They were built to drive and enjoy. Common Sense and Good Judgement goes a long way. 

The stock 400 trans, assuming it is in good condition to begin with, will handle pretty much anything you throw at it, short of nitrous or a blower...

so after some thinking and listening to others, I will be buying a crate engine and new transmission to be put into the corvette.  I will also be keeping the engine that is being pulled out.  I believe it is the original engine.  I have only owned the corvette for one year.  So I do not know what has or has not been done to the engine.  This way I will know all about the engine and will be able to have a warranty with it.  Thank you all for your advice.

Good plan.  👍

Good luck, I ordered a 355, 300 hp crate from Jegs back in April. Was told end of October. Hope you have better luck. AKT I believe do the crates for a lot of the sellers and they say they just don’t have any blocks and parts to build for awhile. 

You might try getting a crate engine from Chevrolet as they have several performance ratings and a 3-year warranty.

I've also heard good things about Blueprint Engines.