I don't send TTM, so whatever I say doesn't affect the outcome one way or another. I like to get as many different things signed as I can of the players I like or the sets I like.
For example, and I would certainly not send them all at once, but if I thought I could get a players entire career card run signed over time, I might try. I think 1-3 is a fair amount and perhaps an offer to keep "extras" is always a good idea. Many of the players probably wouldn't, but some might. I have read stories where guys sent customs and the players really liked and wanted some. That would be the perfect way to thank a player for signing for free.
I have a Claude Osteen complete Topps run from 1959-1976 signed, including a few extras like the 74T Traded and 72T IA cards. Do I need a couple dozen Osteen autos? No. Do I enjoy having each different year signed to compare and contrast? Definitely!
EDIT: After reading all the replies, I did want to say that there should be no reason for anyone to feel bad about trying to sell extra autographs obtained from players either IP or TTM. I have read a number of stories where collectors would request large numbers of signatures from good players in order to have trade material. Often those trades could have turned into sales. Trading only removes a step. If you get 6 cards signed and trade 4 for other cards, you essentially put those into the market and got something in return. Not much different than selling in my mind.
These players have a unique opportunity to create something special with a flick of the wrist. They could very easily capitalize on it and use the money for personal gain or charities, which some do. Nothing wrong with that either, although you have to question some of the larger fees, especially on common players. Where else can someone sign their name and basically create money by doing it? That is pretty cool. I believe that if I were famous enough to warrant demand for my signature, I would sign graciously and not sweat what happened to them. If you are like Bob Feller or Bobby Doerr, you bring joy to many and in turn actually devalue your signature through heavy demand. If you are like Sandy Koufax or Mike Marshall, you artificially inflate the price and demand and surprise...it creates a frenzy that just might "turn off" a guy from signing. Is it really that surprising to these guys? It shouldn't be!
The great guys are the ones who sign freely and don't care what happens to the signatures, because they can always create more.