This thread compelled me to look back at my list of favourite "Star Trek" episodes and in doing so, I found that only two don't have guest stars, if we define 'guest star' as any actor in the episode that isn't a regular cast member (not including one shot minor crew members).
1. The Trouble with Tribbles (Season 2, Episode 13)
2. Space Speed (Season 1, Episode 24)
3. Balance of Terror (Season 1, Episode 8)
4. Journey to Babel (Season 2, Episode 15)
5. The City on the Edge of Forever (Season 1, Episode 28)
6. The Naked Time (Season 1, Episode 6)
7. The Enemy Within (Season 1, Episode 4)
8. Mirror, Mirror (Season 2, Episode 10)
9. This Side of Paradise (Season 1, Episode 25)
10. Day of the Dove (Season 3, Episode 11)
Of that list, I think only "The Naked Time" and "The Enemy Within" are 'regular cast' episodes.