"Big" groups $G$ with trivial $Out(G)$ We are looking for examples of groups $G$ such that $G$ is "big", but $Out(G)$ is trivial. By "big" we mean things like virtually free, or large, or Golod-Shafarevich. However, we would like our groups to be residually-finite.
Edit: Henry, Igor and Lee thank you for all your help. Eventually, we only needed finite $Out(G)$ with $G$ having a trivial center. Thus, $A*B$ was the example we used. It is included in "Large normal subgroup growth and large characteristic subgroup growth" (sorry for the self-promotion, but I thought people might be interested in our motivation and the results of their help). 
 A: A countably-infinite family of groups is given at the end of Section 6 of a paper of A. D. Logan1: for each natural number $n>1$, the two-generator-, one-relator group
$$G_n := \langle a, b; (a^{−2}ba^4ba^{−3}ba^5b)^n\rangle$$
has trivial outer automorphism group.
EDIT This has nothing to do with the above, but I don't want to add a third answer :) My discussion with Henry in the comments to the OP, might suggest that $Out(PGL(2, \mathbb{Z}))$ might be trivial. Apparently, this was proved by Hua and Reiner in 1951-1952, and then disproved(!) by Joan Dyer in 1978, where she constructed an outer automorphism (known ever since as the Dyer automorphism, so $Out(PGL(2, \mathbb{Z})) = C_2;$ the "so" is not trivial but true.
For completeness, her automorphism $\mathcal{D}$ is described as follows. The generators of $PGL(2, \mathbb{Z})$ are:
$$S = \pm \begin{pmatrix}0 &1\\ -1 & 0\end{pmatrix}, T=\pm \begin{pmatrix}1 &1\\ 0 & 1\end{pmatrix},B=\pm \begin{pmatrix}0 &1\\ 1 & 0\end{pmatrix}.$$
Then the automorphism sends $S, T, B$ to $SB, TB, B.$
1A. D. Logan: The outer automorphism groups of two-generator, one-relator groups with torsion. Proc. Amer. Math. Soc. 144 (2016) 4135-4150
A: All fundamental groups of finite volume hyperbolic manifolds (of dimension $\geq 3$) have finite outer automorphism groups, and most have trivial outer automorphism groups.
A: There are many, many examples of large, residually finite groups with trivial outer automorphism group. Indeed, any nonabelian virtually special group is large and residually finite. Furthermore, if they're hyperbolic,  Paulin showed that they have finite outer automorphism group unless they split over a virtually cyclic subgroup; in particular, most such examples have trivial outer automorphism group. Instances of this include: 


*

*fundamental groups of closed hyperbolic 3-manifolds;

*random groups at density less than 1/6;

*most small-cancellation groups;

*groups constructed using the Rips construction;
etc etc.
