(Sources: IMDB and TV'S MASH: The Ultimate Guide Book. The episode numbers are rough and take IMDB as a starting point, which counts them a bit differently, so I tweaked them a bit. The book has a "Missing in Action" section that details the times when the major characters (not just the Major characters) did not appear.)
1) Alan Alda (251): Hawkeye was the only one who showed up in every single episode.
2) Loretta Swit (236): IMDB says she was in all of them, but the book tells us of 15 episodes in which she did not actually appear.
3) Jamie Farr (216): The IMDB count may be off, but it seems right he's at #3 since he was in all seasons.
4) William Christopher (213): See Farr.
5) Gary Burghoff (181): Appeared in most episodes of the first 8 seasons.
6) Harry Morgan (180): According to the book, Potter was in all episodes of Seasons 4-11 except solo episode "Hawkeye. Swit was credited, too, but did not appear, so here's an example of her actual appearances being much lower than Alda's.
7) Mike Farrell: (179): Same state line as Morgan, but note that Morgan appeared in Season 3 as a different character, giving him an edge.
8) David Ogden Stiers (131): Appeared in all but one episode in Seasons 6 through 11.
9) Larry Linville: (121): Frank Burns is low on the list for such a memorable character because, while he was in all but 4 episodes while he was on the show, he left after Season 5.
10) Wayne Rogers (72): Appeared in all episodes in the first 3 seasons.
11) MacLean Stevenson (70): In all but two episodes of the show's first 3 seasons. I think IMDB gives Rogers and Stevenson credit for flashbacks, too, in later years.
Notable supporting players including Kellye Nakahara at 160 per IMDB as Nurse Kellye Yamato, though many were uncredited appearances; Jeff Maxwell as Igor (84), Roy Goldman as Roy (71).