Sets A such that A+A contains the largest set [0,1,..,t]

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.

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.