Nesting of a sequence of functions

Composition[f, g, g, f, g]

might be what you are looking for.

Composition[f, g, g, f, g][x]

yields f[g[g[f[g[x]]]]].

Nest[f@*g, f@g@##, 2] &[]

f[g[f[g[f[g[]]]]]]

Nest[f@*g, f@g@##, 2] &[x]

f[g[f[g[f[g[x]]]]]]

Also

flist = {f, g};
Nest[Last[flist = RotateLeft[flist]]@# &, g[], 5]

f[g[f[g[f[g[]]]]]]

and

(Composition @@ ConstantArray[f@*g, 3])[]

f[g[f[g[f[g[]]]]]]