Suppose we have two sequences $C_i$ and $f_i$, where $C_0=\frac{\pi^2}{3}$ and $f_0=i\pi x$ and with the recurrences $$f_{n+1}(x)=\int \left( \int ( f_n(x) + C_n ) dx \right)dx$$ where the integrals are indefinite and any constant terms are set to 0 and also$$C_{n+1}=\frac{(2\pi i)^{n+4}}{(n+5)!}-\frac1{2\pi i}\int^{2\pi i}_0 f_n(x)dx$$
I know a little about recurrence relations and using generating functions etc. to solve them, but I've never come across one involving one series being a function nor one containing integrals!
How could I solve for a non-recursive formula for $C_n$ and $f_n$?
What Jakobian is pointing out is that the statement
is non-sensical if you consider "constant term" to refer to the constant of integration. It assumes that there is a "special" antiderivative that all others differ from by a constant. But this isn't the case. An arbitrary integrable function $f$ simply has a family of antiderivatives all differing by constants, but without any of them standing above the others. So claiming to set "any constant terms to $0$" begs the question "constant term with respect to what other anti-derivative?"
However there is another interpretation of "constant term" here that is more definitive. Your starting function is a polynomial, and constants are polynomials. And integrals of polynomials are also polynomials, so every $f_n$ will be a polynomial. And polynomials also have "constant terms", which can be set to $0$ to select a unique anti-derivatives from among the many.
However, I will also point out that this interpretation (as is true for any interpretation that would actually make sense) makes working with indefinite integrals completely unnecessary. By this interpretation, all of your "indefinite integrals" are actually the definite integral
$$\int_0^x f(t)\, dt$$
As for solving it, assume that $$f_n(x) = \sum_{k=1}^{n+1} a_{nk}x^k$$ and see what recurrence relations the formulas provide for the coefficients $a_{nk}$.