It is well-known that ZFC with the continuum hypothesis is a $Π^2_1$-conservative extension of ZFC.
General question. What is known about the conservativity of $\mathrm{ZFC}+\varphi$ over $\mathrm{ZFC}$, where $\varphi$ is any one of a variety of axioms that contradict the continuum hypothesis?
More specifically, I am interested in the following cases.
- $\varphi$ equals Martin's maximum
- $\varphi$ equals that $2^{\aleph_0}$ is weakly inaccessible
Note that if $\mathsf{CH}$ fails, then there are reals not in $L$, so even $\Sigma^1_3$-conservativity fails. This is the case with both (1) and (2). But the failure of conservativity is even more serious: If a theory proves the consistency of $\mathsf{ZFC}$, then it is not $\Pi^0_1$-conservative over $\mathsf{ZFC}$ (since $\mathrm{Con}(T)$ is $\Pi^0_1$ for $T$ recursive). On the other hand, already Robinson's $Q$, a very weak fragment of $\mathsf{PA}$, is $\Sigma^0_1$-complete, so failure of $\Pi^0_1$-conservativity is worst possible. Again, this is the case with both (1) and (2).
Martin's maximum is extremely strong, probably the strongest extension of $\mathsf{ZFC}$ that has been seriously investigated. It implies not just the consistency of $\mathsf{ZFC}$, but the existence of inner models with many large cardinals, which is significantly stronger. And its reach goes way beyond that: Once you start with (1), no further set forcing extension can change even the projective theory of the reals, so any such extension of (1) is $\Pi^1_n$-conservative over (1) for all $n$.
If $2^{\aleph_0}$ is inaccessible, then in particular $V\ne L$ and, in $L$, there is a strongly inaccessible cardinal. This is significantly stronger than the consistency of $\mathsf{ZFC}$, see here.
As for your general question, two suggestions: Conservativity is the wrong focus in the absence of $\mathsf{CH}$, as the first paragraph indicates. We instead look for maximal consequences in a precise technical sense. In this regard, my suggestion is studying Woodin's $\mathbb{P}_{max}$-theory, of which there are several excellent expositions (by Paul Larson, for instance) beyond Woodin's own book.
If you are interested in interesting statements without large cardinal impact (so that we do not violate $\Pi^0_1$-conservativity outright), then studying the theory of the reals is a good place to start. The book by Bartoszyński and Judah may help you here, as it discusses in detail many examples of extensions of $\mathsf{ZFC}$ that affect different levels of the projective hierarchy in interesting ways, without requiring the extreme strength or the technical know-how of Martin's maximum.