Here is a nice ad hoc start: The set $S=\lbrace 0, 1, 3, 5, 6, 13, 15, 16, 18, 25, 26, 28, 30, 31 \rbrace$ has $S+S=\lbrace 0,1,2,\cdots,62 \rbrace$ (It is the start of [a sequence in the OEIS][1] with next term 63,  but I'm not sure if that helps). Then $T=S+63S$ has 196 members the largest being $1984$ and $T+T=\lbrace 0,1,2,\cdots,3968 \rbrace$. Now $14=31^{0.63944}$ and $196=3968^{0.63699}$ and further iterating will continue to give examples sparser than $k^{2/3}$. The same trick can be done with any finite example. The densities decrease but only very very slightly.

Observe that the places where $S$ has a jump from $j$ to $2j+1$ are at $0,1,6,31$ (numbers of the form $\frac{5^j-1}{4}$) and that there is a central symmetry for $0,1$ and for  $0,1,3,5,6$ and for $S$. This suggests a treasure hunt for an symmetric extension with largest member $1+5+25+125=156$.

 $V=S \cup \lbrace 63, 64, 70, 71, 78, 85, 86, 92, 93  \rbrace \cup (S+125)$ has $37$ members, the largest being $156$. and $V+V=\lbrace 0,1,2,\cdots,312\rbrace$ This is better than the examples above since $37=312^{0.62875}$. and it can be iterated as above. Maybe an even lower density is possible for $W=V \cup \lbrace ?? \rbrace \cup(V+625)$ where the set in the middle is made of several pairs $q,781-q$. I am not sure if the set in the middle of $V$ is the smallest possible.


  [1]: http://oeis.org/A108337