Certain sets of germs at $+ \infty$ of monotone bijections between neighborhoods of $+ \infty$ form groups under composition. This is the case for germs of functions definable in an o-minimal structure, for certain germs lying in Hardy fields, as well as for more abstract functions defined on fields of formal series, such as transseries. 

In this talk I will describe properties of the resulting ordered groups, and show that they can be studied using valuation-theoretic tools adapted to this non-commutative context.