Kiev, Ukraine, is a little out of the way, but transportation is cheap, there are hostels to welcome you, and its currency, while fluctuating, is generally weak against the dollar. Eastern Europe is rich with history, and Kiev's churches, in particular, are beautiful, to say nothing of the dramatic stories of conquest in the region, from the Habsburgs to the Soviets. Eastern Europeans are proud of their home countries and are happy to share their culture with anyone willing to learn.
Costa Rica is usually considered the must-see travel destination in Central America, but neighboring Nicaragua doesn't always get the love it deserves. The country has a good mix of gorgeous old cities and sweeping rainforests. If you eat local and sleep local, Nicaragua can be very cost-effective, as well.
Turkey is a big backpacking draw for a lot of reasons. It's a massive cultural center, as it's been a gateway between Europe and Asia for centuries. The landscapes and the architecture are both as gorgeous as they are ancient. The food is incredible, and (and here's the big draw for backpackers) extremely affordable.
Bratislava is a stone's throw from Vienna and Prague, but more budget-friendly than either of those towns. Hostels are extremely affordable, and the city's architecture and public works are charming. Local tour guides can tell you firsthand of what's it's like when a country changes hands and all the street names change, though those days are long gone now. Bratislava is a painfully underrated European gem.
Portugal has a lot to offer backpackers. The country's public transit is wonderful, so even if you're not up for a hike, getting around isn't a problem. You could check out a trip up the Atlantic coast for scenery, or if you'd rather stay put, Lisbon is a wonderful city with incredibly cheap hostels.
Lose yourself in lush exotic jungles or head to the sea for world-class snorkeling, surfing, and fishing. Indonesia offers backpackers plenty of activities, complete with budget accommodations and cheap eats. On the island of Bali, visitors can feel like they are on an expensive getaway for a fraction of the budget and with everything they could need within reach.
This is another more active backpacking trip, but it's not as isolated as many others. Peru's rainforests, beaches, and, of its ruins make it a must-see. Since the government has banned solo travel to Macchu Picchu, you'll have the comfort and security of seeing it with a group. (At least, it helps to think of it that way.) Peru has a lot to offer travelers, and it's not as challenging as some other places you might tackle.
Not as dense with cities as some of the countries on the list, Nepal is a bit more of an active trek. Still, the scenery is worth it. You don't have to scale Everest to appreciate what Nepal has to offer—plenty of gorgeous vistas dotted with Hindu and Buddhist temples. This is a little more of a challenge than some other choices, but it's well worth it.
If Nepal is challenging for its sparse population, India is challenging for the opposite reason. India's overpopulation can be overwhelming for people who are new to it, and while every major city has hostels and cheap food, knowing the value of what you're getting for your (admittedly small amount of) money can be tricky. Still, the people and the sights are worth it, and the country is a must-visit.
Thailand has a lot of appeal to a backpacker. From its unique culture and food to the striking Buddhist temples, Thailand is a good destination for Americans looking to experience a truly different way of life. Of course, budgetary concerns are also a great factor, as food and lodging are extremely affordable.