This paper studies the descriptional complexity of (i) sequences over a finite alphabet ; and (ii) subsets of $N$ (the natural numbers). If ${\left(s\left(i\right)\right)}_{i\ge 0}$ is a sequence over a finite alphabet $\Delta$, then we define the $k$-automaticity of $s,A_s^k\left(n\right)$, to be the smallest possible number of states in any deterministic finite automaton that, for all $i$ with $0\le i\le n$, takes $i$ expressed in base $k$ as input and computes $s\left(i\right)$. We give examples of sequences that have high automaticity in all bases $k$ ; for example, we show that the characteristic...