The difficulty of computing discrete logs I understand that in Diffie-Hellman it should be hard to compute $a$ given $g$ and $g^a$. 

In computational Diffie-Hellman, it appears to be hard to compute $(g^{ab})$ from $g^a$ and $g^b$.

As for the field, please consider $\mathbb F_p^*$ for this question.

Is it also difficult to compute the discrete log $w$ of more trivial things, such as $g^w\equiv (g^a+1)^b\pmod p$? It may be assumed that $a,b,g$ and $p$ are given.
I will assume for simplicity that you're talking about the full multiplicative group of $F_p$ instead of a proper subgroup, thus there are no problems with $g^a+1$ (except when $g^a=p-1$ which can be trivially ruled out by comparing to $p$).

The quantity $\log_g(g^a+1)$ is sometimes referred to as the Zech logarithm (strictly speaking, it is defined for characteristic 2, but can be defined here as well). 

So let's define $Z(a)=\log_g(g^a-1)$ and note that it sets up a one-to-one correspondence between $a$ and $Z(a)$. Thus the difficulty of discrete logs is not really changed by introducing the kind of twist you ask about, in your reference to "trivial things".

I can't dig out the reference now, but there was a paper years ago by a French author, in a crypto related conference or journal, on whether precomputing Zech's logarithms on some subset of $F_p^{\ast}$ helped with discrete logarithms. The answer was, "not so much".