So I learned about this interesting invariant from a colloquium by Burglind Jöricke.
(Warning — this post concerns math I don’t know well and is all questions, no answers.)
Suppose you have a holomorphic map from C^* to M_g,n, the moduli space of curves. Then you get a map on fundamental groups from 
But not just any mapping class; this one, which we’ll call u, is the monodromy of a holomorphic family of marked curves around a degenerate point. So, for example, the image of u on homology has to be potentially unipotent. I’m not sure (but I presume others know) which mapping classes u can arise in this way; does some power of u have to be a product of commuting Dehn twists, or is that too much to ask?
In any event, there are lots of mapping classes which you are not going to see. Let m be your favorite one. Now you can still represent m by a smooth loop in M_g,n. And you can deform this loop to be a real-analytic function
Finally — while you can’t extend f to all of C^*, you can extend it to some annulus with outer radius R and inner radius r.
Definition: The conformal modulus of a mapping class x is the supremum, over all such f and all annuli, of (1/2 pi) log(R/r).
So you can think of this as some kind of measurement of “how complicated of a path do you have to draw on M_{g,n} in order to represent x?” The modulus is infinite exactly when the mapping class is represented by a holomorphic degeneration. In particular, I imagine that a pseudo-Anosov mapping class must have finite conformal modulus. That is: positive entropy (aka dilatation) implies finite conformal modulus. Which leads Jöricke to ask: what is the relation more generally between conformal modulus and (log of) dilatation? When (g,n) = (0,3) she has shown that the two are inverse to each other. In this case, the group is more or less PSL_2(Z) so it’s not so surprising that any two measures of complexity are tightly bound together.
Actually, I should be honest and say that Jöricke raised this only for g = 0, so maybe there’s some reason it’s a bad idea to go beyond braids; but the question still seems to me to make sense. For that matter, one could even ask the same question with M_g replaced by A_g, right? What is the conformal modulus of a symplectic matrix which is not potentially unipotent? Is it always tightly related to the size of the largest eigenvalue?
Quomodocumque 
One of my topologist readers who doesn’t like commenting tells me that the monodromy of a map from C^* has to be have the property that some finite power is a product of disjoint Dehn twists.
I think Royden’s Theorem (Teichmueller metric = Kobayashi metric) answers a couple of these questions. My constants here are probably off…
A holomorphic map of an annulus (with finite modulus) lifts to a periodic
holomorphic map of the hyperbolic plane H^2 into Teichmueller space. The
action of Z on H^2 is generated by a hyperbolic isometry f, while the
action on Teichmuller space is generated by the mapping class u determined by the annulus.
According to Royden’s Theorem, the map of H^2 (with curvature -4) is a
contraction, and so the translation length L(f) on H^2 is at least the
translation length L(u) on Teichmueller space. On the other hand, the
modulus of the annulus H^2/ is exactly pi/(2L(f)) (maybe the 2 is in
the wrong place… I get confused with curvature -4). Therefore, the
modulus pi/(2L(f)) is at most pi/(2L(u)).
Now, Bers’ proof of Thurston’s classification theorem says that L(u) = 0
if and only if u has a power which is a product of commuting Dehn twists.
Therefore infinite modulus implies L(u) = 0, and hence u is a product of
commuting Dehn twists.
On the other hand, if u is pseudo-Anosov, then L(u) = ent(u), the entropy
of u. So the modulus is at most pi/(2 ent(u)). The Teichmueller disk
defined by the axis of u gives an isometric periodic embedding of H^2 into
Teichmueller space, making the inequalities into equalities, so that
modulus = pi/(2ent(u)).
If u is reducible, it’s not clear to me whether you can find a sequence of
annuli for which the supremum of the associated moduli is exactly
pi/(2L(u)).