This is a very difficult question to answer, and will depend on many factors. There is no final answer. However, many people have affirmatively declared that $15,000 (US) is the amount you'll need. Others have said much less, while some say it's much more.
Ultimately, you'll need to estimate your approximate daily spending, and then do some math (more on that later, though).
There are too many factors that affect a travel budget, so there is no way to simply give a rough estimate of cost. This is true of any trip, whether it is a few days or several months or more. What type of hotels are you staying in? How will you get from place to place? Transportation and accommodation are two of the most expensive areas that travelers spend money on, but there are many more.
Where you choose to visit will have the biggest impact on your budget. Europe and the U.S. are expensive, while India and Cambodia are dirt cheap. Naturally, spending a longer period of time in cheaper locations will allow you to stretch your budget and travel longer. There are some surprisingly expensive destinations that you should be aware of. Brazil and Argentina are much more expensive than some of their nearby South American neighbors. West Africa is also quite expensive due to limited hotel and restaurant options (those that are available are overpriced and of low quality). Eastern Europe remains cheaper than most of Western Europe, but prices continue to rise and Eastern Europe is no longer the extreme budget destination it once was. This is particularly true for places like Greece and Croatia, that have long been discovered by travelers and vacationers alike. Asia is a mixed bag, with more developed countries costing more, and many developing countries (particularly in Southeast Asia) being much cheaper.
Your travel style can have a significant effect on your budget. If you're on a truly tight budget then you will likely have to sacrifice some level of comfort and luxury during your trip. Using public transportation is cheaper than hiring taxis or renting cars, but often not as efficient or relaxing. Staying in hostels or budget hotels is probably the most effective way to save money. You'll need to figure out your priorities, though. Are you willing to share a bathroom, or even sleep in a shared room in order to travel for a longer time period? But with these sacrifices come some rewards. In many places, locals take public transportation so you'll have a chance to interact with the people of the community and you'll get to experience what daily life is really like. Although hostels offer less privacy, they also give you the chance to meet other travelers, many of whom are also on long term trips. These travelers are an excellent source of information and travel tips. It's also fun to have the company, particularly if you've been traveling by yourself for a while.
The time of year you travel to some places can also make a difference in cost. Traveling in the low season or shoulder season can often land you significant discounts. These discounts are not nearly as large as the savings you will get from staying in hostels or budget hotels, however. Also, be aware that some travel destinations essentially close down during the low season. This is particularly true in areas with extreme weather fluctuations. If most of the area's hotels and restaurants are closed, then those few that remain open will likely raise their price significantly. Also, if you're traveling somewhere specifically to experience the outdoor activities, then make sure you go during the best season. Places like Nepal or Patagonia are obvious trekking destinations. There are very specific seasons when it is best to travel there, and you are better off if you arrange your trip around these times, but the prices are also highest during these periods.
How quickly you travel from place to place will significantly affect your budget as well. Transportation is expensive, especially if you fly often. Taking local overland transportation tends to be the cheapest way to go, but it's not as cheap as just staying put for awhile. Besides, taking a longer time to get to know a place is when you open yourself up to the most interesting experiences. Another benefit to traveling slowly is that it gives you time to simply walk around a place. If you're only in town for a few days, then you'll likely try to visit all the museums and attractions that you can within those few days. This will dramatically increase your daily budget and, in turn, raise the total cost of your trip. Take your time and soak up the local atmosphere if you're on a tight budget. You never know what you might discover, as you will have time to get off of the beaten path.
Tours and guides are another expensive part of traveling. Often you can see the same places on your own and learn from a guide book instead of a person. While the quality of the information you might learn from a book versus a person is debatable, there's no arguing that some tours can be extremely expensive. Many tours also include transportation and meals, so it may be worthwhile to consider this before making a final decision. If you do decide to take a tour, ask other travelers who they would recommend. Personal recommendations mean a lot more than any suggestion in a guide book or on the internet.
Children will add considerable expenses to your budget. You'll probably want to travel with a little more comfort and stability if you have kids. That's understandable. Don't let that stop you though! Your entire family will benefit from the experience.
Many travelers often get caught up in the "game" of saving money while traveling. Don't fall into this trap. If you plan ahead and know your budget, then you can take advantage of many of the opportunities that will come your way. Don't try to justify every cent you spend. Just focus on enjoying yourself.