Sets A such that A+A contains the largest set [0,1,..,t] I look for a reference for the following problem.
Given an integer $k$, find a set $A\subset\mathbb{N}$ with $|A|=k$
that maximizes $t$ such that $\left[0,1,..,t\right]\subset A+A$.
 A: This is related to ``thin additive bases" of order $2$.   Clearly $t$ cannot be larger than $k(k+1)/2$.   It is also possible to give examples where $t$ grows quadratically.  Take $A=A_0 \cup A_1$ where $A_0$ contains all integers below 
$t$ with binary expansion $\sum_{j} \epsilon_j 2^j$ with $\epsilon_j= 0$ unless $j$ is even, and $A_1$ consists of numbers with binary digits $\epsilon_j=0$ unless $j$ is odd.  Then $A$ has $O(\sqrt{t})$ elements in it; or alternatively $t\ge Ck^2$ for some constant $C>0$.   See for example this paper of Blomer which has other references. 
A: A table of values for these $t$ are given in the introduction Graham and Sloane's On Additive Bases and Harmonius Graphs (your sequence corresponds to $n_\beta(k)$ in their notation).  Graham and Sloane also give some references to previous work with this sequence, both under the name of "interval basis" (or Abschnittsbasis), going back to a paper in German from Rohrbach in the 1930's, and under the name of "The Postage Stamp Problem".  
This is sequence A001212 in the OEIS, which has additional references.  
