An ordinary die is thrown until the running total of the throws first exceeds 12.

Claim: If you throw a die until the running total exceeds n>=5, a final
outcome of n+1 is more likely than any other.

Assume we throw an m for a total n+k>n+1, and assume m-k>=0.  Now, it
is just as likely to throw an m as an m-k+1, which means that the sum
n+1 is just as likely as any other.  Now consider the series of throws
consisting of n-5 1's followed by a 6 and note that we cannot achieve
more than an n+1 by changing the last die roll.  Hence, a total of n+1
is more likely than any other.