Oh!!! So many countries, so little time. With unlimited funds and a year to travel, I would first go to Colombia for about two months to visit people I know and enjoy the warm weather and some of my favorite foods that I can't get here or have a hard time making (pan de bono, yucca bread, coconut ice cream, chontaduros, etc.).
I would go through Ecuador just because I've never been there, and head for Peru. There I would visit friends near Chiclayo and stay for a month or two. From there, I'd go to Trujillo, spend a few days, and then on to Lima for a week or so. Then I'd FINALLY, finally visit Machu Picchu.
After that, I'd go to Argentina for a week, and then Brasil for a while. I don't know how long I'd stay there, because I've always wanted to visit that country.
Off to Europe. I'd probably start with Scotland and Ireland, then France, Spain, Italy, Greece, and wherever God leads me.
Then I'd go to Ukraine. I'd visit Kiev and the Crimean peninsula. I'd probably stay in Simferopol for a week or two, and visit a few friends. Then I'd go to Moscow.
Lastly, I'd go to Japan and China, maybe India, perhaps Egypt.
Anybody willing to host me? ;)