Rolls of a five-sided die to reach a total of five

A fair die with faces $1,2,3,4,5$ is rolled again and again until the running total is $5$ or more. What is the expected number of rolls?

$1296/625$. For an $n$-sided die and target $n$, the chance that $k$ rolls have not yet reached $n$ is $\binom{n-1}{k}/n^k$ (compositions of a number below $n$ into $k$ parts). Summing over $k\ge0$ gives $E=\left(1+\tfrac1n\right)^{n-1}$, which is $(6/5)^4$ for $n=5$.