Let $X$ be a smooth compact 4-manifold. 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?



Behavior of genus function on a 4-manifold for sums
In the case that $x\cdot x \neq 0$, topological methods based on the G-signature show that the genus goes to infinity more or less quadratically in $n$. (I'll be more specific below.) This goes back to Rochlin (Two-dimensional submanifolds of four-dimensional manifolds) and Hsiang-Szczarba (On embedding surfaces in 4-manifolds) 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
$$
g \geq \left|\frac{(h^2-1)(\xi \cdot \xi)- \sigma(X)}{4 h^2}\right| - \frac{b\_2(X)}{2}.
$$
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 G-signature bound. 


Work of Strle (Bounds on genus and geometric intersections from cylindrical end moduli spaces) gives stronger results for surfaces of positive self-intersection in the case that $b\_2^+(X) =1$, without the assumption of non-vanishing Seiberg-Witten invariants. See also recent work of Konno (Bounds on genus and configurations of embedded surfaces in 4-manifolds).


Finally, in the case of self-intersection $0$, the growth is at most linear (and possibly $0$, as Marco notes). This follows by tubing together parallel copies of a given surface.


