Oh it all depends on what you´d like to see, what kind of roads you want to ride on and what bike you´d like to rent I´d say. And WHEN you´d like to do the trip. I don´t know much about rental costs these days but here are some tips about regions that might be nice for a 2 - 3 weeks of riding.....
* If you want to save money then I´d say Eastern Europe. Romania, Bulgaria, Serbia, Montenegro, Hungary, Slovakia...... Food and hotels doesn´t cost as much as in western Europe and LOTS to see. Don´t know anything about bike rentals tho..... Sorry for that. But Romania alone is well worth two weeks and don´t miss Transfagarasan and Transalpina. These parts won´t work during our winter/your summer tho since there is snow......
* I did a trip some 10 + years around the eastern provinces in Canada + New England and that was a great trip indeed. Rented a bike in Boston, drove up and visited a friend outside Montreal and drove east along the St Lawrence river, through New Brunswick, over to PEI and then to Nova Scotia and after that the ferry to Maine and back to Boston. That trip took me 10 days but could easily have lasted tree weeks! Prince Edward Island is beautiful and northern Nova Scotia is just magnificent! Don´t miss Cape Bretagne and the Cabot Trail! There is a whisky destillery (Glenora) located just south of Inverness. I stayed there for one night. Good fun. The french canadian border guard heading into Quebec was grumpy as hell. I was on a rented Harley with US plates and he spoke only french and gave me a very hard time until I handed him my passport and he realized that was not from USA but from Sweden. Then he excused himself in perfect english was very friendly all of a sudden. He gave me cookies.

Wonderful trip in general but too cold during our winter time as well....
* Another suggestion could be to fly into Munich/southern Germany and rent a bike. I think that Munich has an international airport. But from Munich you have easy access to the Alps, Switzerland, Austria, northern Italy, France, Spain..... That could be a great 2 - 3 wek trip also.
* Norway is beautiful but expensive as hell.
* We went to Iceland last summer. Very diverse and amazingly beautiful and I´m rather certain that they rent out bikes in Reykjavik but 2 - 3 weeks might be too long. We spent one week there and could easily have stayed longer. The island is bigger than one might suspect when looking on a map but three weeks......?
* Alaska/western Canada/north western USA could also be a great trip. Never been there myself tho but that is also on the to-do-list. I´ve friends that has flown into Anchorage and rented bikes and they had a great time. And they drove around for 3 weeks if I don´t recall things wrong.
I don´t know much about areas south of the equator so others could perhaps suggest things there. It all comes down to what part of the year you´d like to do this. Again, sorry but I don´t know much about rental costs these days but eastern Europe and USA/Canada is most likely cheaper than western Europe/Norway/Iceland.....
All the best.
//Jan