What happens when you equip Helm of the Host to Teysa Karlov? Do the Teysas trigger each other making infinite copies of the death trigger? My thinking is that it says: when a triggered ability is caused by death, copy it. Is that ability copied by the other Teysa or is the original ability just copied again? So instead of one Blood Artist trigger, would I get 3? Or would I just win the game doing infinite damage?
You get one more trigger for each Teysa Karlov you have. So if you have 2 Teysa Karlov, you get 3 triggers.
From the Rulings on the Gatherer page:
1/25/2019 — If you somehow control two Teysas, a creature dying causes abilities to trigger three times, not four. A third Teysa causes abilities to trigger four times, a fourth causes abilities to trigger five times, and so on. This also means that if you control Teysa and cast a second one, an ability that triggers when it dies due to the “legend rule” triggers three times.
It works this way because Teysa Karlov's ability is a replacement effect, and replacement effects can only apply to an event once. When a death trigger occurs, one Teysa sees it and replaces it with two death triggers, then the other Teysa sees it and replaces it with three death triggers. Both Teysas have now applied their replacement effects to this one event and can't apply additional times, so they won't loop infinitely.
Teysa's ability isn't triggered by another creature dying, it's triggered by an ability that's triggered by a creature dying. For instance, suppose a creature has "When this creature dies, gain one life". When that creature dies, its ability is triggered, and "gain one life" is put on the stack. Teysa's ability is then triggered. It's triggered not by the creature dying, but by "gain one life" being put on the stack. Teysa will then put another "gain one life" on the stack, but since this was not triggered by a death, it will not trigger Teysa. Teysa is only triggered when a trigger is specifically from a death, not whenever a death is part of the causal chain that leads to the trigger. If you have an additional Teysa, it will see the original "gain one life" being put on the stack, and this will trigger it putting an additional "gain one life" on the stack. So:
"Gain one life" put on stack.
One of the Teysas sees "Gain one life" being put on the stack as a result of a death, and so puts an additional "Gain one life" on the stack. The other Teysa sees "Gain one life" being put on the stack as a result of a death, and so puts an additional "Gain one life" on the stack. The other "Gain one life", however, was not put on the stack from a death, and so does not trigger Teysa.