Thirteen. Consider the next-to-last throw: Its total must be 12, 11, 10, 9, 8, or 7. If it’s 12, then the final total will be 13, 14, 15, 16, 17, or 18, all equally likely; if the next-to-last total is 11, then the final total will be 12, 13, 14, 15, 16, or 17; and so on. The only candidate to appear in every case is 13, so it’s the most likely total.

Generally, if we replace 12 by N, then the answer is N + 1.

(C.C. Carter and N.J. Fine, “The Most Likely Total,” American Mathematical Monthly 55:2 [February 1948], 98.)