AFAIK Foghat has had the most lineup changes of any rock band. Somewhere around 30 different members at different times IIRC. Hell, at one time there was two Foghats touring.
Has any other band even come close to this amount of changes and still kept recording under the samr name?
Uriah Heep had quite the revolving door of drummers and bass players within the span of just a few years back in the early-to-mid-70s. Probably nowhere close to thirty but a good dozen or so at least. They even put a chart on the back of their “Best of…” LP so you could keep track. 
Just as an aside…
I think Uriah Heeps Sweet Freedom Is one of the best albums ever done.
And their Look at Yourself album cover, with the reflective foil, one of the most creative album covers.
One of the “original” doo-wop groups–the Drifters?–can be seen at a zillion different casinos at any one time. There were multiple replacement members, and each started his own group with the same name when he moved on. All the original members are dead and apparently no one cares that the name is being used. I don’t know if any of them are recording under the name.
Well, all of the following people have been members of bands called King Crimson. Only Robert Fripp has been part of every KC lineup:
Adrian Belew
Bill Bruford
Boz Burrell
Mel Collins
David Cross
Robert Fripp
Michael Giles
Peter Giles
Trey Gunn
Gordon Haskell
Greg Lake
Tony Levin
Pat Mastelotto
Ian McDonald
Jamie Muir
Richard Palmer-James
Pete Sinfield
Keith Tippett
Ian Wallace
John Wetton
And I’ve lost track of all the people who’ve been members of Rainbow/Ritchie Blackmore’s Rainbow.
I don’ t know all their names, but I know the Kinks have had quite a number of bandmembers (even more if you consider the brass bands, back up singers, and dancing girls that have been included in their live shows); if I can find the article, at one point in the late 70s, Dave Davies said in an interview he wasn’t exactly sure who was in the band at that point.
Rainbow members:
Ritchie Blackmore
Graham Bonnet
Tony Carey
Roger Glover
Don Airey
Jimmy Bain
Chuck Burgi
Bob Daisley
Ronnie James Dio
Gary Driscoll
Craig Gruber
Paul Morris
Cozy Powell
Bob Rondinelli
David Rosenthal
Micky Lee Soule
Joe Lynn Turner
Doogie White
David Stone
Candice Night
Traffic only once put out two consecutive albums with the same personnel. They don’t reach in absolute numbers, but probably are up there in frequency, with Dave Mason constantly leaving and rejoining.
However, the All Music Guide lists 40 group members for Blood Sweat and Tears, and that doesn’t include the string section that was on their first album.
Spinal Tap:
Nigel Tufnell
David St Hubbins
Derek Smalls
2 other guys whose names I can’t recall
32 drummers
Not that BOC achieved the heights of revolving-door membership that some of the bands here did, but it should be noted that they lost enough members that they derisively came to be known as “Two Oyster Cult.”
To my surprise, the Allman Brothers Band has only had 21 different members (not including guests. That’s still up there, but it sounds like that’s way behind Foghat and others.
NoCoolUserName
Ben E King is still alive and he was a member of the Drifters way back when. (I think he was an original).
Paul Revere and the Raiders have been legendary for personnel changes.
The section of the website www.paulrevereraiders.com hasn’t been updated for quite a while but I think that just in the 1960’s they had about 2 dozen different members as well as people who left and came back. The current touring band has been surprisingly the most stable some of whom have been members for 25 years. However the only original member of the group is … Paul Revere.
And I thought Yes had a lot of members. Hmmm.
Well, the All Music Guide puts Hawkwind as having had 34 members. But if you count Nik Turner’s “solo” work (after Dave Brock fired everyone), you can add probably another dozen or so people.
Okay. I couldn’t resist chasing down the list of Alice Cooper band members - from the original group to the present. Now, I know people might want to disqualify Alice as a solo artist with studio musicians, but he’s really not that different from many other “bands” (including some that have already been mentioned here), and he does have a live band backing him at shows.
Before we get to the list, note that I’ve excluded pre-“Alice Cooper” (1968) members John Speer (drums) and John Tatum (guitar), extra vocalists, and studio-only musicians. The two most notable studio-only musicians would have to be producer Bob Ezrin (keyboards 1970-2001) and bassist Tony Levin (1975-1977). (And some of the years listed may be off.) Anyhoo, here goes…
ORIGINAL BAND [5]
Alice Cooper (vocals 1964-)
Michael Bruce (guitar/keyboards 1966-1974)
Glen Buxton (guitar 1964-1974)
Dennis Dunaway (bass 1964-1974)
Neal Smith (drums 1967-1974)
GUITARISTS [16]
Mick Mashbir (1973-1974)
Dick Wagner (1975-1983)
Steve Hunter (1975-1979)
Jefferson Kewley (1978)
Davey Johnstone (1979-1980)
Mike Pinera (1980-1982)
John Nittzinger (1981-1982)
Devlin 7/Johnny Dime (1986-1988)
Kane Roberts (1986-1989)
Al Pitrelli (1989-1990)
Pete Freisen (1989-2002)
Stef Burns (1991-1998)
Vinne Moore (1992)
Reb Beach (1996-1999)
Ryan Roxie (1996-)
Eric Dover (2002-)
BASSISTS [9]
Prakash John (1975-1983)
Erik Scott (1980-1983)
Kip Winger (1986-1987)
Todd Jensen (1986-1999)
Steve Steele (1988)
Tommy “T-Bone” Caradonna (1989-1990)
Greg Smith (1991-2002)
Chuck Wright (2001-2002)
Chuck Garric (2003-)
DRUMMERS [9]
Whitey “Penti” Glan (1975-1979)
Ross Salamone (1980)
Jan Uvena (1981-1983)
Ken Mary (1986-1988)
Jonathon Mover (1989)
Eric Singer (1990-2004)
Jimmy Degrasso (1996-2000)
Winston Watson (1998)
Tommy Clufetos (2004-)
KEYBOARDISTS [9]
Bob Dolin (1973-1974)
Jozef Chirowski (1975)
Fred Mandel (1977-1980)
Duane Hitchings (1980-1982)
Wayne Cook (1982)
Paul “Horrorvitz” Taylor (1986-1998)
Derek Sherinian (1989-1999)
Lyndsey Vannoy (1998)
Teddy “Zigzag” Andreadis (2000-2003)
Indents! Neat! So, let’s add this up. Five original members and 16 other guitarists makes 21. Three times nine for bassists, drummers and keyboard players makes 27. Add that up for a total of… 48! So far. 
Not too bad. (An even 50 if we include Speer and Tatum.)
Anyhoo…
You will note that only rarely will groups include members that are famous in their own right. Of course, members from different bands sometimes team together for form “supergroups”, but rarely does an existing band take on members of other famous bands.
An exception is Dashboard Confessional, which, while still being partially a Chris Carrabba vehicle, has Sunny Day Real Estate’s Dan Hoerner as a member, according to allmusic.com. In addition, and I don’t know if he’s an official member or not, Scott Schoenbeck, formerly of The Promise Ring provides the bass.
Of course, rarely does this opportunity occur, since established artists don’t want to be seen as trying to fill someone’s shoes. But in this case, Carrabba was trying to expand the focus of the band from a mainly acoustic-ish sound to a full-fledged modern rock sound, so took on the new members as part of the new sound.