There is no limit. If you have more than one out, each one will trigger independently and give you a life.
The only three limits are:
- Legendary permanents with the same name,
- Planeswalker permanents sharing a subtype and
- Enchantments with the World supertype.
704.5j If two or more planeswalkers that share a planeswalker type are on the battlefield, all are put into their owners’ graveyards. This is called the “planeswalker uniqueness rule.”
704.5k If two or more legendary permanents with the same name are on the battlefield, all are put into their owners’ graveyards. This is called the “legend rule.” If only one of those permanents is legendary, this rule doesn’t apply.
704.5m If two or more permanents have the supertype world, all except the one that has been a permanent with the world supertype on the battlefield for the shortest amount of time are put into their owners’ graveyards. In the event of a tie for the shortest amount of time, all are put into their owners’ graveyards. This is called the “world rule.”
The correct answer, as any deckbuilding Johnny will tell you, is always "as many as possible". In Magic it's quite easy to come up with a deck that will put infinite artifacts into play:
4 Seat of the Synod 4 Tree of Tales 4 Ancient Den 4 Great Furnace 4 Vault of Whispers 4 Island 4 Kuldotha Forgemaster 4 Vedalken Engineer 1 Darksteel Colossus 4 Kaleidostone 4 Everflowing Chalice 4 Brittle Effigy 4 Mycosynth Lattice 4 Summoning Station 4 Thornbite Staff 2 Akroma's Memorial 1 Legacy Weapon
See this article for a full explanation of how to get infinite Pincher tokens (Mycosynth Lattice-d so they are also artifacts) into play, but I include an excerpt that should give you the idea:
With a Staff'ed Forgemaster, sacrifice three other artifacts to activate the Forgemaster. (They can be anything besides the combo-related cards.) Summoning Station will untap thrice, making three artifact Pincher tokens. Meanwhile, your newly untapped Forgemaster (because of the Staff) will have fetched a Darksteel Colossus. Now repeat this process, except you'll sacrifice two of those Pinchers and the Colossus itself. Since the Colossus shuffles itself back into your library, you can find it again (or Legacy Weapon.) In this way we won't run out of artifacts to keep the engine running.
Eventually we'll have infinite Pincher tokens. When you're ready to swing, fetch up an Akroma's Memorial and go to town with Pinchers of Wrath!