Here's an example where a sequence of manifolds approximates one given manifold in a definite
sense.  

For $i\in {\Bbb N}$, let $M_i$ be an orientable surface of genus $i$.

Let $T$ be the boundary of a tubular neighborhood in ${\Bbb R}^3$ of 
$$\{ (x,0,0)_{x\ge 0} \} \cup \{ (x,1,0)_{x\ge 0}\}\cup \{ (i,y,0)_{i\in{\Bbb N},y\in[0,1]}\}\ .$$

Now one can have maps from $T$ to each $M_i$, each map a homeomorphism 
when restricted to an ever larger open set, in such a way that these
open sets exhaust $T$.

In general I suppose you'd want a net of manifolds of a fixed dimension and a corresponding
directed set of open sets that exhaust the manifold you mean to approximate.  You wouldn't even have to have the limit manifold in advance if you had a suitable family of compatible maps connecting the various manifolds in your net.

I doubt you can, in any reasonable way, approximate the topology of a compact topological manifold by topological manifolds of the same dimension.