Shouldn't silver be unreactive? It forms a #+1# cation doesn't it?

Not necessarily.

In isolation, as an atom that just acquired a full octet, yes. A full octet makes an atom more stable and lower in energy, making it happier.

In the context of a fluid solution, no. It can certainly react with some other ion(s), depending on the ion(s).

In a solution, for example...

#Ag^(+)# has a "full charge" (really, "full octet"), with an electron configuration of #[Kr]4d^(10)5s^0# when a #+1# cation.

It is not unreactive in solution with #Cl^(-)#, with an electron configuration of #[Ne]3s^2 3p^6#, another atom that has acquired a full octet and become a #-1# anion.

#Ag^(+)# wants to precipitate with #Cl^(-)# in aqueous solution. That means it wants to form a new solid compound. The solubility product constant #K_"sp"# makes that favorable.

#AgCl(s) rightleftharpoons Ag^(+)(aq) + Cl^(-)(aq)#

#K_"sp" ~~ 1.8xx10^(-10) M^2 = [Ag^(+)][Cl^(-)]#

This means that not much can successfully dissociate into the aqueous solution, and so that means most of it favorably becomes solid #AgCl#. That means what just happened was the reverse of the reaction written above.

(Therefore, it just reacted.)