In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A general setup for such a result in algebraic geometry is:

Given a proper, generically smooth map $\pi:X \rightarrow S$ of relative dimension d, say S is connected. This gives rise to an $l$-adic representations of the etale fundamental group $\pi_1(U)$ where $U$ is smooth locus of $\pi$ corresponding to higher pushforward $R^d \pi_* Q_l$. One might say it has "big monodromy" if the Zariski closure of the image is as big as it can be given that it has to respect cup-product, etc.

My specific question is what are the geometric consequences of big monodromy? If we know such a result for $\pi$, what does that say about the geometry of the fibration or at the very least is there geometric intuition for what it should mean?

I welcome intuition from number theory, algebraic geometry, or complex geometry.

I have also heard that "one should expect big monodromy unless there is a reason not to" (for example, complex multiplication). What are other examples of things which inhibit big monodromy?