Yeah, Guest, in a nursing home you can often honor the same request repeatedly and no one will care.  But in a residence for higher-functioning seniors, there is often one person who has begun 'slipping' who will repeatedly request the same song.  The rest of the folks DON'T want to hear it over and over. But you can usually appease the requester by saying "Well, we already did that one.  How about [some other favorite] instead?"  Usually they'll let you move on.

Don M.,
I think you made an important point.  If the gig is one where you're being paid to sing what the audience wants (e.g., at a private party or at a club that features a certain type of music), trying to comply with requests is part of the job.
It's a real problem, though, when you've got a couple of folks in the audience who want to dominate the requests and/or whose requests are out of sync with the tastes of the bulk of the audience.

What's really frustrating is when I have  carefully constructed a set with just the right mix of songs in just the right sequence, and then sabotage all my work by blurting out requests, uninvited.  E.g., one song leads naturally into  patter that merges into the next song, or a series of songs builds to an emotional or musical crescendo--and the requests throw the rhythm of the whole program off.  In a normal adult concert setting, this is seldom a problem, [so that's usually where I do this type of program] but it often is a problem in bars or with children or folks in nursing or assisted living facilities.

I almost never refuse a request because I'm tired of a song.  I've done enough theater to know that part of your art is keeping the same thing fresh night after night.  It's especially hard with comedy, but that's the job.
The payoff is not in the material per se, which is old to you, but in the connection with the audience, which is different every night.

It's kinda like sex.  Some people care more about their own favorite moves and positions, regardless of what their partner likes.  Others [who probably have better sex lives] get off on pleasing their partners as well as themselves.  If you find a position [song] that turn you both on, it's fantastic!

The one kind of song I usually refuse to sing [solo], though, is one that just isn't suited to my voice or gender.  Some songs just don't sound right to me when sung by a [straight] woman.  ["Lay, Lady, Lay" comes to mind.]  Others need a belting-type voice [e.g., Janis Joplin's "Take Another Little Piece Of My Heart" or some sea chanteys].  Or someone requests a band song, though I'm solo; can you imagine one woman with an acoustic guitar making "Black Water" sound at all palatable?  [Using the sex analogy, I feel that on some songs, even if I comply with the request, neither my 'partner' nor I will be satisfied with the outcome.]

leprechaun, the requests to stop singing could be lucrative, if folks would fold 'em up inside $20 bills!

Mark C.,
Thanks for the Bill Monroe line ["We don't do that particular number but here's one that has a lot of the same notes."].  I'll have to 'borrow' that one for times when I can't do the one requested.

Harnish, does "some stuff acoustic with no pa" mean "bastard acoustic stuff?"