Using Nest to build continued fraction

This is what you need:

Nest[x/(1 + #) &, x, 3]

Try the following.

f = Replace[#, 1/a_ -> x/a] &;

g[n_Integer] := Map[f, Nest[1/(1 + #) &, x, n], Infinity] // f;

Let us check

g[5]

g[10]

Have fun!