G = Exp[-(x-y)^2/(t_1-t_2)]/(x+y+t_1+t_2)
i1[t1_?NumericQ, t2_ ?NumericQ] :=  i1[t1, t2] = NIntegrate[G, {x,-∞, ∞}, {y, -∞,∞}]
i2[t1_ ?NumericQ]:= i2[t1] = NIntegrate[i1[t1,t2],{t2,0,t1}]
NIntegrate[i2[t1],{t1,0,∞}]