The fundamental problems of Coding Theory are provably hard.

Berlekamp, E.R., R.J. McEliece, and H.C.A. Van Tilborg, 1978, "On The Inherent Intractability Of Certain Coding Problems", IEEE Trans. Inform. Theory, IT 24:384-386.

That paper very soon led McEliece to design a cryptographic scheme which is provably hard to break.

McEliece, R.J., "A Public-Key Cryptosystem Based on Algebraic Coding Theory", JPL DSN Progress Report 42-44, pp. 114-116, Jan-Feb 1978.

The game of LR Hackenbush is NP.

Pages 211-217 of Winning Ways: Volume 1 second edition, A K Peters, Ltd., January 2003.

Loony Dots-and-Boxes endgames are NP.

Pages 577-578 of Winning Ways: Volume 3 second edition, A K Peters, Ltd., August 2003.

Some games with very simple orthodox strategies have complex canonical forms.

Pages 26-28 of Sums of Nx2 Amazons, by Elwyn Berlekamp, Institute of Mathematics Statistics "Lecture Notes -- Monograph Series", 35:1-34, 2000.