I was testing a deck playing The Gitrog Monster on MTGO and was surprised when, discarding 2 land cards from my hand during cleanup step due to having 9 cards in hand, Gitrog triggered twice. I knew that Gitrog could trigger during cleanup step, and that it would keep creating more cleanup steps as long as I kept discarding a land to make my hand too large, but I did not think discarding multiple cards during cleanup would trigger multiple Girog triggers since Gitrog says "one or more". Is this the intended interaction, or is MTGO broken again?
You discard excess cards during the cleanup step simultaneously, not one at a time (under C.R. 514.1, a player with excess cards as the cleanup step begins "discards enough cards" to reduce the hand size as required), so The Gitrog Monster's last ability should have triggered at most once, even if two or more land cards are discarded this way (C.R. 603.2, especially C.R. 603.2c; C.R. 700.1). The issue you found with Magic Online is not new; see this thread.
EDIT (Sep. 23): Added rule citation.