There is actually a card limit (archived+open), its just some huge number, as mentioned here : Trello Limits
Your limits may vary, the link just confirms it exists. As well as goes through all the other limits and an example value to ballpark it.
To check what your board limit is, go to Menu>More>Print and Export>Export as JSON. Then search for "totalperboard" with ctrl+F. The first result will be the board limit for total cards, the latter will be for lists. A random board of mine for instance has:
So Trello will only warn me once I hit a combined 1.8 million cards (archived and not) on a single board. And will disable new cards at 2 million.
You may have a different if equally mind-boggling large limit :P
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