  We present the first examples of elements in the fundamental group of the
space of Legendrian links in the standard contact 3-sphere whose action on the
Legendrian contact DGA is of infinite order. This allows us to construct the
first families of Legendrian links that can be shown to admit infinitely many
Lagrangian fillings by Floer-theoretic techniques. These families include the
first known Legendrian links with infinitely many fillings that are not rainbow
closures of positive braids, and the smallest Legendrian link with infinitely
many fillings known to date. We discuss how to use our examples to construct
other links with infinitely many fillings, in particular giving the first
Floer-theoretic proof that Legendrian (n,m) torus links have infinitely many
Lagrangian fillings, if n is greater than 3 and m greater than 6, or
(n,m)=(4,4),(4,5). In addition, for any given higher genus, we construct a
Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category
can distinguish infinitely many exact closed Lagrangian surfaces of that genus.
A key technical ingredient behind our results is a new combinatorial formula
for decomposable cobordism maps between Legendrian contact DGAs with integer
(group ring) coefficients.
