A problem by Kwesi Levy

Let $p$ be any positive prime number. What is the length of the longest repetend* required to express $0.1_{10}$ in base- $p$ ?

*The "repetend" is the repeating block of a non-terminating, rational decimal. For example, in base-10, $\frac{1}{3} = 0.\overline{3}$ has a repetend of length 1 while $\frac{1}{990} = 0.0\overline{01}$ has a repetend of length 2 (we ignore the leading decimal[s]).