This reaction is less trivial than one would expect.
At first, you get a precipitate. The NH3 forms NH4(+) and OH(-). The OH(-) reacts with the Ag(+).
When more NH3 is added, then a complex is formed. Ag(+) ions form a fairly stable complex with NH3, this complex is [Ag(NH3)2](+). So, any precipitate formed redissolves again, but now the Ag(+) is locked up in the complex.
When the solution is allowed to stand for a long time, and when the concentration is sufficiently high (or the pH is suffciently high), then the complex [Ag(NH3)2](+) decomposes, giving highly complicated macroscopic structures, with Ag(+), coordinated to the nitrogens of the ammonia, but with hydrogen atoms replaced by Ag. A complicated non-stoichiometirc compound is formed, which sometimes is called "silver nitride", but that name by no means precisely describes the nature of this compound. Anyways, the compound formed in this solution is VERY explosive and for this reason one should not store solutions with the [Ag(NH3)2](+) ion in it.