The biggest number

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.

This famous math activity usually limits the time period to 10-20 seconds, but longer periods can be fun, too. However, OP forgot a rule: Your answer must define a specific integer in such a way that an educated mathematician can, given infinite time, reliably arrive at your integer with only the information on your card. That rules out answers like "infinity", "Jack's answer + 1", or "the biggest number ever imagined + 1".

Graham's Number
In order to use Graham's Number, you would have to define it, because it has no universal variable. If you have to define it, you might as well use something better. See below.

BB(BB(...BB(999)...))
While nested Busy Beavers seems a natural extension, no one has yet (to my knowledge/googling) published a serious peer-reviewed paper doing so. Meaning you've got nothing to cite if someone wants to challenge you on it.

I think, given a very tight time limit, the best solution would be something like the following, getting in as many 1's as possible:

BB(1111111111111111111111)

This is bigger than BB(1111) ↑↑↑↑ BB(1111) but took the same amount of time to write. 1's are faster to write than 9's and ↑'s took me about twice as long to write as 1's. If you've got a minute or longer, I would suggest defining a Graham-esque number utilizing Busy Beavers. Something like the following, perhaps:

Answer = a_BB(1111),
where a₁ = BB(1111) ↑↑↑↑ BB(1111),
a_n = B(1111) ↑^a(n)-1 BB(1111)

Basically Graham's number, substituting each 3 with BB(1111), done with BB(1111) iterations rather than 64. After you've got that done, if you've got more time, write in some more ones, preferably in the first line.