Here's my research on the live-action Godzilla franchise. Suggestions, changes, explanations, commentary welcome. I'll change the list according to your remarks.
1. Godzilla (1954)*
2. Godzilla Raids Again (aka "Gigantis the Fire Monster," 1955)
3. King Kong vs. Godzilla (1962)*
4. Mothra vs. Godzilla (aka "Godzilla vs. The Thing," 1964)*
5. Ghidorah, the Three-Headed Monster (1964)*
6. Invasion of the Astro-Monster (aka "Monster Zero," 1965)*
7. Ebirah, Horror of the Deep (aka "Godzilla vs. the Sea Monster," 1966)
8. Son of Godzilla (1967)
9. Destroy All Monsters (1968)*
10. All Monsters Attack (aka "Godzilla's Revenge," 1969)
11. Godzilla vs. Hedorah (aka "Godzilla vs. The Smog Monster," 1971)
12. Godzilla vs. Gigan (aka "Godzilla on Monster Island," 1972)
13. Godzilla vs. Megalon (1973)
14. Godzilla vs. Mechagodzilla (aka "Godzilla vs. The Bionic Monster," 1974)
15. Terror of Mechagodzilla (aka "The Terror of Godzilla," 1975)
16. The Return of Godzilla (1984)
17. Godzilla vs. Biollante (1989)
18. Godzilla vs. King Ghidorah (1991)
19. Godzilla vs. Mothra (aka "Godzilla and Mothra: The Battle for Earth," 1992)*
20. Godzilla vs. Mechagodzilla II (1993)
21. Godzilla vs. SpaceGodzilla (1994)
22. Godzilla vs. Destoroyah (1995)
23. Godzilla (1998)
24. Godzilla 2000: Millennium (1999)*
25. Godzilla vs. Megaguirus (2000)
26. Godzilla, Mothra and King Ghidorah: Giant Monsters All-Out Attack (2001)*
27. Godzilla Against Mechagodzilla (2002)
28. Godzilla: Tokyo S.O.S. (2003)
29. Godzilla: Final Wars (2004)*
30. Shin Godzilla (2016)*
31. Godzilla (2014)
32. Godzilla: King of the Monsters (2019)*
33. Godzilla vs. Kong (2021)
*Recommended by Baron Bizarre
1. Gamera the Giant Monster (1965)
2. Gamera vs. Barugon (1966)
3. Gamera vs. Gyaos (1967)
4. Gamera vs. Viras (1968)
5. Gamera vs. Guiron (1969)
6. Gamera vs. Jiger (1970)
7. Gamera vs. Zigra (1971)
8. Gamera Super Monster (1980)
9. Gamera Guardian of the Universe (1995)
10. Gamera 2: Attack of Legion (1996)
11. Gamera 3: Revenge of Iris (1999)
12. Gamera the Brave (2006)
Looks good to me, Skipper.
"Godzilla vs. Mothra" (1992) was also known as "Godzilla and Mothra: The Battle for Earth"
Many of the Showa Era pictures had alternate names that you may see them mentioned under:
2. Godzilla Raids Again (1955) "Gigantis the Fire Monster"
4. Mothra vs. Godzilla (1964) "Godzilla vs. The Thing"
6. Invasion of the Astro-Monster (1965) "Monster Zero"
7. Ebirah, Horror of the Deep (1966) "Godzilla vs. The Sea Monster"
10. All Monsters Attack (1969) "Godzilla's Revenge"
11. Godzilla vs. Hedorah (1971) "Godzilla vs. The Smog Monster"
12. Godzilla vs. Gigan (1972) "Godzilla On Monster Island"
14. Godzilla vs. Mechagodzilla (1974) "Godzilla vs. The Bionic Monster" and "Godzilla vs. the Cosmic Monster"
15. Terror of Mechagodzilla (1975) "The Terror of Godzilla"
Updated with your info, Bob!
Pieces of a few - not my genre.
Godzilla: Saw both versions of the original, the one without Raymond Burr* just recently. Saw a couple of the others, including the one where G starts out as a tiny lizard that crawls on its belly, which strains my suspension of disbelief. Giant monsters just don’t do it for me.
* When I first saw Hitchcock's Rear Window I was surprised that Raymond Burr was playing a heavy. It turns out that in the early part of his career he was always playing a bad guy.
Shouldn't all the Toho kaiju films be included?
Hit me! I’ll add ‘em!
Did I miss if someone has proposed another daikaiju series for inclusion, one that involves a turtle who is a friend to children?
How many can I think of?
Varan the Unbelievable (1958)
Dogora (1964) aka Dagorah the Space Monster
Frankenstein Conquers the World (1965)
The War of the Gargantuas (1966)
King Kong Escapes (1967)
Yog - Monster from Space (1970)
Rebirth of Mothra (1996)
Rebirth of Mothra II (1997)
Rebirth of Mothra III (1998)
How many did I miss?
RE: shared universe in the Toho kaiju films:
Varan the Unbelievable introduces Modern Mysteries, the in-universe show that reappears in Ghidorah, the Three-Headed Monster, the film (as far as I can tell) that establishes the fact of a shared universe for most of the Toho kaiju films. (Yes Godzilla had met Mothra earlier the same year, but we're still uncertain about the others being in one universe). Even the outsize Frankenstein movies get folded in later through Baragon's appearance in Destroy All Monsters, and I'm sure some other connections may have been made by the post-Showa era films, but I am mostly unfamiliar with those.
A few of these (Yog/Space Amoeba, for example) have no established connection to the others, so far as I can determine, save for the studio and the house style. Someone may know more about this point.