The dilation, centered at $2 + 3i,$ with scale factor 3, takes $-1 - i$ to which complex number?
Let $z$ be the image of $-1 - i$ under the dilation.

[asy]
unitsize(0.5 cm);

pair C, P, Q;

C = (2,3);
P = (-1,-1);
Q = interp(C,P,3);
draw((-10,0)--(10,0));
draw((0,-10)--(0,10));
draw(C--Q,dashed);

dot("$2 + 3i$", (2,3), NE);
dot("$-1 - i$", (-1,-1), NW);
dot("$-7 - 9i$", (-7,-9), SW);
[/asy]

Since the dilation is centered at $2 + 3i,$ with scale factor 3,
\[z - (2 + 3i) = 3((-1 - i) - (2 + 3i)).\]Solving, we find $z = \boxed{-7 - 9i}.$
Let $z$ be the image of $-1 - i$ under the dilation.

[asy]
unitsize(0.5 cm);

pair C, P, Q;

C = (2,3);
P = (-1,-1);
Q = interp(C,P,3);
draw((-10,0)--(10,0));
draw((0,-10)--(0,10));
draw(C--Q,dashed);

dot("$2 + 3i$", (2,3), NE);
dot("$-1 - i$", (-1,-1), NW);
dot("$-7 - 9i$", (-7,-9), SW);
[/asy]

Since the dilation is centered at $2 + 3i,$ with scale factor 3,
\[z - (2 + 3i) = 3((-1 - i) - (2 + 3i)).\]Solving, we find $z = \boxed{-7 - 9i}.$
The final answer is $$-7 - 9i$$. I hope it is correct.