Example of pairwise independent random process with expected max load $\sqrt{n}$
Here's how to do it.

First, choose a random $k$ between 1 and $n$ to be the "crowded bin".
Next, choose a random permutation $\pi$ of $1,2,\ldots, n-1$.

Now, for $1 \leq i \leq n-1$,
$$ \mbox{put ball } i \mbox{ into bin } 
\begin{cases} k \ \ \ \ \ \ \ \ \ \ \ \ \mbox{with probability }\ \frac{1}{\sqrt{n}}, \\
k + \pi(i) \mbox{ with probability }1 - \frac{1}{\sqrt{n}},
\end{cases}$$
where the sum is taken mod $n$.  

Put ball $n$ into a random bin.

Consider balls $i,j$ where $1 \leq i &lt; j \leq n-1$. The bins these two balls fall into are independent. They both go into the crowded bin with probability $\frac{1}{n}$, and the crowded bin is equally likely to be any of the $n$ bins. And if they don't both go into the crowded bin, by symmetry they are equally likely to be in any two different bins. Thus, the bins these two balls are placed into are independent.

Since ball $n$ is placed into a random bin, it is independent of the other balls. 

The expected max load here is essentially the expected number of balls in the crowded bin, which is
$$\frac{n-1}{\sqrt{n}}+\frac{1}{n} = \sqrt{n} + o(1).$$