It is easy to see that for a finite set of integers $A$ of cardinality $n$, the cardinality of the sumset $A+A$ satisfies $$ 2n-1\leq |A+A|\leq \frac{n(n+1)}{2}. $$

The lower bound is essentially attained when the set $A$ is an arithmetic progression while the upper bound is attained for sets such that $a_{i}+a_{j}=a_{p}+a_{q}$ implies that $\{i,j\}=\{p,q\}$. An example of this set is

$$ A=\{2^{i}:i={0,1,\ldots,n-1}\}. $$

I think is a fun problem (probably not very difficult) to study what is the rate of growth of $|A_{p}+A_{p}|$ as $n\to\infty$ where $$ A_p=\{1^p,2^p,\ldots,n^p\}. $$

Define the "asymptotic growth exponent" as $$ \mathrm{ge_{p}} = \lim_{n\to\infty} {\frac{\log(|A_{p}+A_{p}|)}{\log(n)}}. $$

What is the limit $\mathrm{ge_{p}}$ for $p=2$? In general?

Enjoy! :-)