I have given you all an index card. You have one minute to write down the biggest number that you can (barring infinity) using any mathematical function that you know (addition, subtraction, exponentation, ackermann, Knuth's up arrow, etc). Go.
I thought the point of BB was that if you ran a program on a computer of complexity N, and it didn't stop after BB(N) steps, then it would never stop. Therefore you could write a program to find a counterexample to Goldbach's conjecture, and if it didn't find one after Y billion years, then you'd know there wasn't any.