1) Nunn's ChessBase puzzle 7: Construct a legal game that ends with (the black move) 5...Rh1 mate. Note that the last move is not a capture.
2) Related puzzle: Construct a legal game that opens 1.e4 and ends with (the black move) 5...NxR (knight takes rook) mate.
I like to have at least one diagram in my entries whenever natural so here is a nut where the final position is the main key:
3) Construct a legal game that ends with this position after EXACTLY four moves (that is eight half moves, four by White and four by Black).
This kind of short games where you are supposed to reconstruct the entire game by some sparse hints are often called "Shortest proof games". I am not sure that the diagram challenge really belongs in this category, as it is definitely no challenge to reach this position in three moves (six half moves) e.g.: 1.e4 e6 2.Bc4 c6 3.Bxe6 dxe6.