  In this paper we introduce the notions of topological entropy and topological
pressure for non-autonomous iterated function systems (or NAIFSs for short) on
countably infinite alphabets. NAIFSs differ from the usual (autonomous)
iterated function systems, they are given \cite{LGMU} by a sequence of
collections of continuous maps on a compact topological space, where maps are
allowed to vary between iterations. Several basic properties of topological
pressure and topological entropy of NAIFSs are provided. Especially, we
generalize the classical Bowen's result to NAIFSs ensures that the topological
entropy is concentrated on the set of nonwandering points. Then, we define the
notion of specification property, under which, the NAIFSs have positive
topological entropy and all points are entropy points. In particular, each
NAIFS with the specification property is topologically chaotic. Additionally,
the $\ast$-expansive property for NAIFSs is introduced. We will prove that the
topological pressure of any continuous potential can be computed as a limit at
a definite size scale whenever the NAIFS satisfies the $\ast$-expansive
property. Finally, we study the NAIFSs induced by expanding maps. We prove that
these NAIFSs having the specification and $\ast$-expansive properties.
