Is there a limit to the amount of armor you can have?

There is no practical limit to the amount of armor you may possess, because while armor can increase without bound on Warriors (and to a lesser extent, Druids), the game will never last long enough to get anywhere close to any programmatic limit (due to Fatigue, which cumulatively increases each time it hits you).

According to the wiki, even if you survive 49 turns over fatigue (by which time you would have been out of cards for at least ~22 turns), the game ends in a draw on the 50th turn.

So Armor Up, and sleep soundly knowing that more armor is never wasted!

With 6 Armorsmiths, a Pyromancer on one side, 5 Armorsmith, a Pyromancer and a Lorewalker Cho on the other, each side can play 10 Circles of Healing a turnto gain 42 and 36 armor each. Each round includes both players' turns so one would gain 84 armor and the other 72.

At 50 rounds max, the minimum fatigue is 23? I estimate about 1000 armor to be the maximum achievable.