Which actor has appeared the most number of times as themselves in a
movie or television show?

Alfred Hitchcock (who wasn't an actor of course) usually made some
minor appearance in his movies.

This doesn't help at all but I can't think of ANYONE off hand.

Go on, somebody, surprise me!

Ah yes ... I now recall that, during WWII, lots of actors appeared as
themselves in 'Stage Door Canteen' and 'Thank Your Lucky Stars'.

Of course, it just has to be Ronald Reagan!

Do you mean like, Lucile Ball in her own show?

Do you want to include Dean Martin who appeared as himself (in his own
television show) for 245 episodes?

How about Johnny Carson who was originally an actor and then appeared
as himself for 30 years on The Tonight Show?

Or have I totally violated the spirit of this question?

Have a good day.

Oh Klemo

Lucille (note the spelling) Ball usually appeared as Lucy Ricardo in her own show.

I thought EVERYBODY knew that!

PB

I stand corrected on both counts.
Research has proved Bryan correct
KEMLO






PS I'm not old enough to remember the earlier shows

Another thing to consider is at what point an actor who appears too
many times as  him/herself becomes a 'television personality' instead
of an actor.

This is a very interesting question in the spirit in which it's
intended. I haven't the foggiest idea what the answer is.

   But it is food for thought. If one defines "appears as self"
broadly enough, then it's possible to say that any actor who is cast
on its box office appeal is appearing "as self", since the box office
draw is based upon the perceived attractiveness of the actor as a
personality, rather than on the character being played.

   Thus, Adam Sandler, Arnold Schwartzenegger, John Wayne, etc., all
appear as themselves in features which are marketed on them as
name-brand stars.

Could it possibly be the actor Adam West? According to this Wikipedia
page on him, after appearing in the role of batman on TV, he has
appeared as himself in movies and television, most recently as Mayor
Adam West on the animated show "The Family Guy"!

http://en.wikipedia.org/wiki/Adam_West

Jay

John Wayne.   Did he ever play a role where the character was any
different from his own, including, of course, his last one about
fighting the big C?

YES 
He was the centurian in the film 
The Greatest Story Ever Told

Just as an interesting aside, legend has it that Wayne was
falling-down drunk when he was delivering his lines as the centurian.

It depends on the definition of "actor" and the definition of "television show".
Do you include people like David Letterman who have acted in a couple
of tv shows/film(s), but has also appeared hundreds of times in his
own TV show? - Though he's acted he's better known as a host than an
actor.

Do you include "actors" who have only supplied the voice to a character?
Do you include all talk shows, etc. or just fictional shows.

Here's my sort of "Top 10 list" made by searching the Internet Movie
Database.  It's not totally accurate as it doesn't include the
hundreds of times people like David Letterman have hosted their show. 
Maybe Bob Hope could be the answer as he is better known as an actor.

top 10 list
Stern, Howard             tot films for him/her self=8 tot progs=1793
Quivers, Robin            tot films for him/her self=5 tot progs=1109
DeGeneres, Ellen          tot films for him/her self=45 tot progs=446
Parton, Dolly             tot films for him/her self=125 tot progs=277
Hope, Bob                 tot films for him/her self=190 tot progs=176
Garnel, Felipa            tot films for him/her self=0 tot progs=290
Pires, Paulo (I)          tot films for him/her self=1 tot progs=224
Madonna (I)               tot films for him/her self=86 tot progs=133
Probst, Jeff (I)          tot films for him/her self=7 tot progs=209
Friis-Mikkelsen, Jarl     tot films for him/her self=14 tot progs=198

Here's another one! Besides making numerous appearances on
entertainment shows, Wladziu Valentino Liberace (best known as just
Liberace) made a habit of popping up as "himself" on soap operas,
situation comedies, night-time dramas... even Wrestlemania!

Here's a link to his filmography   http://www.imdb.com/name/nm0508766/

Jay