A number so large that writing down its digit count would itself take forever
Ronald Graham needed an upper bound for a question in Ramsey theory about colouring the edges of a hypercube, and the number he came up with is so vast that the observable universe couldn't hold its digits even if every digit were a single Planck volume. Given a number with that unimaginably many digits, can you say anything at all about it, such as what its very last digit is?
Reveal the answer
The last digit is 7 (the last ten digits are ...2464195387), findable through modular arithmetic even though the number as a whole is far too large to ever compute in full. Graham's number was, for a time, the largest number ever used in a serious mathematical proof and entered the Guinness Book of Records on that basis; the actual answer to the Ramsey problem it bounds is now known to be far smaller.