2
Why not just nest the integration?
integrals = NestList[Integrate[#, {x, 0, x}] &, Sin[x], 5]
Plot[integrals, {x, -1, 1}, PlotLegends -> "Expressions"]
You can then get the function $f_n(x)$ for a given $n$ with integrals[[n]]
Since you mentioned Manipulate, you can do this and control parameters like a and the number of integration levels....
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