I found the following position online where White's best course is to force a threefold repetition with the sequence Qg5-Kh8-Qf6-Kg8 because all other moves lose:
[Title ""]
[FEN "q4rk1/5p1p/5Q2/8/8/6P1/p4P1P/6K1 w - - 0 1"]
[startflipped ""]
1. Qg5 Kh8 2. Qf6 Kg8
This example uses 11 pieces and 29 material points (i.e., Queen = 9, Rook = 5, Bishop = 3, Knight = 3, Pawn = 1). I'm wondering if a position can be found with fewer pieces and/or fewer material points.
4 pieces is the minimum, and is attained by this position though Black has the option of trading Q for N to a bare-king draw:
[Title "Draw by perpetual check or insufficient material"]
[FEN "8/8/8/8/8/7q/4NK2/7k w - - 0 0"]
1. Ng3+ Kh2 2. Nf1+ Kh1 3. Ng3+ Kh2 4. Nf1+ Kh1 5. Ng3+
For material points, this 5-piece position has only 5 MP's, and the perpetual check is absolutely forced
[Title "Draw by perpetual check"]
[FEN "8/8/8/8/8/7p/p3NK2/7k w - - 0 0"]
1. Ng3+ Kh2 2. Nf1+ Kh1 3. Ng3+ Kh2 4. Nf1+ Kh1 5. Ng3+
[FEN "8/8/8/8/8/7p/8/5K1k w - - 0 1"]
In this position, white's best is 1.Kf2 to force threefold repetition with 1...Kh2 2.Kf1 Kh1 etc.
Of course black can deviate, but that's also a draw (1...h2 2.Kf1 stalemate, or 2...Kg3 3.Kg1 h2+ 4.Kh1 Kh3 stalemate).
