I am a very curious person, and I was experimenting in-game with Scaled Nightmare. If you don't know, Scaled Nightmare is a 2/8 dragon that doubles its attack every turn, the highest I've gotten being around 1000 or so damage. So, I was wondering, what would be the highest possible amount of damage any one minion could have?
2147483647 (so 2.1 billion; maximum of a 32 bit signed integer) After that it loops back to 0.
This was tested a long time ago with health as seen in this video. Attack behaves the same way.
Should Blizzard ever choose to increase the variable to 64-bit, the maximum would be 2^63 = 9.22 x 10^18 = 9.22 Quadrillion