@user21820 If you need it for the CLT (and I haven't seen a proof without it) then it is difficult to imagine studying things like entropy and energy distribution in thermodynamics without it. Ironically, the same author here gives several proofs of it, if you're curious to see if you can escape it. The first two proofs he gives make use of it. I haven't gone through all of them:
https://terrytao.wordpress.com/2010/01/05/254a-notes-2-the-central-limit-theorem/