Well, I know this is a touchy subject. I had a similar situation with my DD, only I don't think it was quite as much as yours. But I began to feel a difference in attitude with DD. I never did mind nursing her, but when she began to DEMAND to nurse, I instinctively did not want to nurse her at those moments. It was definitely her attitude!
After much deliberation, I decided to set some limits with her at about age 18 months. What I did was pick the easiest time to start with. For DD, that was mornings. I decided that she could nurse first thing in the morning, and then nurse again at nap time. And during this first "phase" she could also nurse whenever she wanted outside of that time frame. I also made sure she got a good breakfast and a mini-meal just before her nap time so I wouldn't be worried if she was truly hungry.
I included lots of play time and cuddle time too, and did some of her favorite activities with her during that time to help distract her. Whenever she made a request for "na na" I would say "yes, you may have na na at naptime."
I decided to wait until she was comfortable with that before I began with more limits. And it took about 6 weeks for her to really stop asking for nursing during that morning time. But, what WAS nice was that she already knew what my answer would be ahead of time, and so there was no meltdown when I said "later."
After several weeks, I started working on the afternoons. I decided she could nurse after she awoke from her nap, and then had to go 1/2 way til bedtime (around 3-4pm) and then she could have a nursing session. Again, outside of her morning routine and the newly established afternoon routine, she could nurse whenever she wanted.
Now, we've got it worked out that she nurses: first thing in the morning, nurses down for a nap, occasionally nurses upon waking from a nap, nurses once in the afternoon, and at bedtime. I also nurse her if she's hurt herself. She's also recently gone through her first illness, and I have nursed her freely and as often as I could through the illness, as she really needed the comfort, nourishment, and hydration.
She still asks for more, but she knows what my answer will be, and I think that is key to avoiding meltdowns.