Hierarchy theorem for circuit size
In fact it is possible to show that, for every $f$ sufficiently small (less than $2^n/n$), there are functions computable by circuits of size $f(n)$ but not by circuits of size $f(n)-O(1)$, or even $f(n)-1$, depending on the type of gates that you allow.

Here is a simple argument that shows that there are functions computable in size $f(n)$ but not size$ f(n)-O(n)$.

We know that:


there is a function $g$ that requires circuit complexity at least $2^n/O(n)$, and, in particular, circuit complexity more than $f(n)$.
the function $z$ such that $z(x)=0$ for every input $x$ is computable by a constant-size circuit.
if two functions $g_1$ and $g_2$ differ only in one input, then their circuit complexity differs by at most $O(n)$


Suppose that $g$ is nonzero on $N$ inputs. Call such inputs $x_1,\ldots,x_N$. We can consider, for each $i$, the function $g_i(x)$ which is the indicator function of the set $\{ x_1,\ldots,x_i \}$; thus $g_0=0$ and $g_N=g$.

Clearly there is some $i$ such that $g_{i+1}$ has circuit complexity more than $f(n)$ and $g_i$ has circuit complexity less than $f(n)$. But then $g_{i}$ has circuit complexity less than $f(n)$ but more than $f(n) - O(n)$.