Since the commenters are not turning their comments to an answer, I will summarize:
If the observable universe is made of around 10^80 hydrogen atoms*, this implies each atom should be exchanging gauge bosons with 10^80 other atoms.
So far so good. The field theoretical view of interactions says that these happen by the exchange of virtual particles between "real particles", real means that they are on their mass shell.
This Feynman diagram
Shows the on shell particles: The kaon which decays to three pions, which have definite mass, and off shell ones, the quarks and the gluon and the W which are off mass shell and virtual and thus do not obey energy and momentum conservation laws. They are just quantum number representations similar to the putative on shell particles ( gluons and quarks are never free, so never on shell, but that is another story).
So each atom should launch 10^80 gravitons, and a similar number of photons, gluons etc.t
No. Each atom theoretically can interact with every other atom through virtual particle exchanges, though these interactions will be extremely extremely weak the longer the distance between the on shell particles. These off shell particles may be one or many, from each hydrogen atom, though in the calculation usually a few exchanges are dominant.
Launch implies on shell manifestation, in my vocabulary.
What you should keep is that there is no conservation on the number of off the mass shell virtual particles other than the diminishing probability of multiple exchanges, diminishing at least for weak, electromagnetic and gravitational interaction. Gluons are another type of animal and another story too.