Is there any episode which is more well-known? Unlikely. And for good reason. This episode contains a little of everything that made Star Trek good and memorable.
The story is quite unique, and makes wonderful use of the little universe beginning to develop within the show. Every character (save Sulu) gets a mini feature, with those by Scotty and Uhura being truly memorable. There are also several memorable guest stars including the under-utilized William Campbell. All of our heroes get to relax a bit and have some fun. The recently dour McCoy gets some funny lines, and the Vulcan gets to exercise his human half a bit.
We see some very useful scenes (missing from recent episodes) of routine life aboard a starship. From Kirk's headache, to his chicken soup and coffee, to Uhura relaxing with friends around a table, to Scotty and Chekov sparring verbally over liquor, we get glimpses of everyday life which help us relate to the characters in the long run. The episode's lone sour note comes during one such scene -- the Spock/McCoy scene in sickbay -- that is redeemed by the routine nature of the activity taking place. Luckily, the great dislike that Gene Coon tried to cement between these two characters would never take hold, and (after a fashion) ultimately mellowed nicely into occasional light sparring.
Though easy to overlook, there are several nice moments when the music maintains the tone of the episode. This episode features several long sequences without dialogue, and the music becomes essentially an extra guest star. The most prominent of these is when the bartender removes tribbles and places them on the bar as Jones looks on. But note that there are numerous sequences where Kirk looks around at the tribbles and the music reveals his thoughts.
There are some close calls with the plot (why would the Federation leave such important grain unguarded in the first place? how could Darvin have wandered around a station filled with tribbles and not been discovered earlier? who would open a grain container door while standing beneath it?), these are rather easy to forgive because the overall charm of the episode makes it irresistible.
(On a side note, when DS9 returned to this episode nearly 30 years later, they created their own immensely charming take on the story. It's a rare instance where a later series successfully added subtext to an existing episode/character. Watching the scene of Kirk being pelted by the randomly falling tribbles has a new undercurrent of humor knowing that Sisko and Dax are doing the throwing...)
Rating: Very Top (1)