I really should avoid answering questions late at night. My original answer is muddled enough to not work. But here is what it should have been:
Let $X_0 = \{0,1\}$ and $X_i = X_{i-1} \times X_0$. Give each $X_i$ the $2-$adic ultrametric. That is the distance between two sequences of $0'$s and $1'$s is $2^{-n}$ where $n$ is the number of share initial symbols the two sequences share. The projections maps are given by truncation. The inverse limit is now $\mathbb{Z}_2$ which has box counting dimension equal to one.