The Trouble with Tribbles (Star Trek: The Original Series, Season 2, Episode 15)
Captain Kirk finds himself and the crew up to their eyeballs in adorable (but dangerous!) little floof balls. Possibly the greatest episode of any show ever. It scored three Emmy nominations and a follow-up episode in Star Trek: Deep Space Nine's "Trials and Tribble-ations."
Watch Star Trek: The Original Series on CBS All Access.