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.
Loony Dots-and-Boxes endgames are NP.
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.