Let $X$ be a smooth compact 4manifold. Then every element of $H_2(X;\mathbb{Z})$ can be represented by a smooth embedded orientable surface and we have the so called genus function $G: H_2(X; \mathbb{Z}) \to \mathbb{Z}_{\geq 0}$ which assigns to a homology class the smallest genus of such a smooth surface needed to represent it. Suppose that $x$ is a nontorision element of $H_2(X; \mathbb{Z})$. Does the sequence $G(x), G(2x), G(3x),...$ limit to infinity? Can there be arbitrarily large zeroes? Is there always a limit?
2 Answers
In the case that $x\cdot x \neq 0$, topological methods based on the Gsignature show that the genus goes to infinity more or less quadratically in $n$. (I'll be more specific below.) This goes back to Rochlin (Twodimensional submanifolds of fourdimensional manifolds) and HsiangSzczarba (On embedding surfaces in 4manifolds) in the 1970s.
Following Rochlin's version (since I don't have the other at hand): if a homology class $\xi$ is divisible by $h$, an odd prime power, then $$ g \geq \left\frac{(h^21)(\xi \cdot \xi) \sigma(X)}{4 h^2}\right  \frac{b_2(X)}{2}. $$ Writing $\xi = h \alpha$ we see that the right side grows quadratically in such $h$. (Generally this grows as the square of the largest prime power dividing $n$ where $\xi = n \alpha$; presumably the growth rate of that quantity in $n$ is known.)
By looking in a neighborhood (and sticking to prime powers), you can see that you'd expect quadratic growth, but the estimate above looks off by a factor of two. For instance, when it holds, the adjunction formula (as quoted by Marco above) gives a bound that is roughly twice the Gsignature bound.
Work of Strle (Bounds on genus and geometric intersections from cylindrical end moduli spaces) gives stronger results for surfaces of positive selfintersection in the case that $b_2^+(X) =1$, without the assumption of nonvanishing SeibergWitten invariants. See also recent work of Konno (Bounds on genus and configurations of embedded surfaces in 4manifolds).
Finally, in the case of selfintersection $0$, the growth is at most linear (and possibly $0$, as Marco notes). This follows by tubing together parallel copies of a given surface.

$\begingroup$ Thank you very much  this was very helpful. In the case where the self intersection is 0 and the sequence is not constant (as in the answer of @MarcGolla), is it true that the sequence limits to infinity? $\endgroup$ Aug 27, 2018 at 18:42

$\begingroup$ @user101010 It's a reasonable guess. In fact it seems like a reasonable guess that if the sequence is not constant = 0 then it grows without bound. But I don't see how to say anything general on the subject. $\endgroup$ Aug 27, 2018 at 19:40
Sometimes the function $G$ can be constantly 0: consider the class $x = [S^2\times\{p\}]$ in $H_2(S^2\times F)$, where $F$ is a surface. Then $G(nx)$ can be realised by an embedded sphere for all $n$: just pick $n$ distinct points $p_1,\dots,p_n$ in $F$, and tube $S^2\times\{p_i\}$ to $S^2\times\{p_{i+1}\}$ (using pairwise disjoint tubes).
As for the existence of a limit, to me this is a lot less clear. Certainly something is known when $b^+(X) > 1$ and some Seiberg–Witten invariant of $X$ does not vanish, at least in the case when $x\cdot x > 0$. Then there is the adjunction inequality (Kronheimer–Mrowka), telling you that (for some second cohomology class $K$, corresponding to a nonvanishing SW invariant) $$ 2G(nx)  2 \ge \langle K, x\rangle + n^2x\cdot x. $$ The righthand side of the inequality grows quadratically, so $G(nx)$ goes to $\infty$.
I'd be very curious to know of "interesting" behaviours of the function $n \mapsto G(nx)$ (e.g. nonmonotonicity, frequent nonmonotonicity, eventual constant nonzero behaviour, periodicity/aperiodicity).