How BIG
 

I know the Bianchi Pista and Fuji Track come with 700x23 tires, does anyone know how large a tire can be squeezed on?

700 x 23.000001.

but thats just a guess.

I'm running 700 x 26, and could go comfortably wider.
I've seen 700 x 28 for sale, they should fit easily, but beyond that I don't know.
You could go out to 700 x 32 or 35, but you may have to massage (unhook) your brakes to squeeze the tire through. Maybe even let out some air to gain the necessary room to get through the brakes.
It can be done, it could be a pain though.

Generally speaking, wider tires roll slower, but give a less jarring ride.