It seems to me all the answers so far make it look far more complicated than what it actually is.
What I've done is just travel as tourist staying in each country around 3 months.
When I "lived" in Chile I just took weekend 'visa vacations' to Argentina and had my stay extended for another 90 days.
Nothing changed in terms of taxes for me, since I never really become resident of another country.
Earning money online definitely makes things easier since you never have to be paid as a local in whatever country you are at, therefore, not working there illegally or anything like that.
I am not sure how much money your online business makes you, but I usually aim for developing countries, where you can easily live on US $800 - $1200 tops a month. Like bradleyjoyce said I would recommend Latin America. I've lived/traveled/worked in Colombia, Chile, Peru, Argentina, Mexico.
Also, very important, make sure you have someone reliable back home that can help you deal with administrative/misc., stuff like depositing a check, receiving important mail, faxing stuff, and the sort of things you may not be able to do online.
This is great advice. I've done something similar and found that after about 2 months you can get a flavour of what life in that country would be like. It's a fantastic thing to do, and it will completely change your perspective on life in the UK!
It's incredibly important to have someone you can trust to take care of admin issues at home if you're gone a while though. My parents helped me out, but a lawyer or accountant will also work - but cost you a fortune. Possibly you can hire a temporary admin assistant, but make sure you trust that person to look after everything while you're gone; you never know what will happen.
The only problem that I've found has been getting a reliable internet connection for a couple of months at a time - if you have to work on sensitive things or you want to work on things for a long time, you don't want to work from public hotspots. I'd love to get some advice for that situation myself.
You make a deal with the a really nice internet cafe with a good connection. All the kids speaking in a foreign language going crazy over video games might be distracting (especially when school comes out o_O), but it's like a rentable AC office with a good connection and cheap drinks and snacks.
Or can rent a apartment that comes with it's own landline internet connection. The landlords can set it up in their name for you and it would be part of your rent.
Cool. My problem has been finding a room for two months with an internet connection - often there's a minimum period. It's not so bad if you like living with people, but if you live alone...
The idea about the net cafe is a nice one though - maybe I'll try that next time! :-)
This can work in some places, like South America, but not in others, like Europe. In Europe you get a 90-day tourist visa within a 6-month period, so if you stay for 90 days you must leave for and stay out of Europe for another 90 days before you can return to get a new tourist visa.
I can recommend Ecuador. I am currently living on the beach in luxurious apartment with see view, swimming pool, safe gated community , 2Mbit internet connection for $900. All done through AirBnb
Peru is beautiful, but I'm not a fan of Lima's weather (think seattle). So for travel+fun I'd recommend just landing in Lima and going elsewhere in the mountains. If I did not have to recruit engineers, meet entrepreneurs or do biz dev, I'd def. stay in Cusco for a while.
The food and transportation is very affordable, and really really cheap if you don't go to the touristy restaurants and instead head to el mercado (marketplace). You can get a full meal with soup, drink and dessert for like US$.60-$1
Now, if you want to meet fellow hackers/entrepreneurs and recruit incredible talent, check out http://limavalley.com the startup/tech community in Peru.
What I've done is just travel as tourist staying in each country around 3 months.
When I "lived" in Chile I just took weekend 'visa vacations' to Argentina and had my stay extended for another 90 days.
Nothing changed in terms of taxes for me, since I never really become resident of another country.
Earning money online definitely makes things easier since you never have to be paid as a local in whatever country you are at, therefore, not working there illegally or anything like that.
I am not sure how much money your online business makes you, but I usually aim for developing countries, where you can easily live on US $800 - $1200 tops a month. Like bradleyjoyce said I would recommend Latin America. I've lived/traveled/worked in Colombia, Chile, Peru, Argentina, Mexico.
Also, very important, make sure you have someone reliable back home that can help you deal with administrative/misc., stuff like depositing a check, receiving important mail, faxing stuff, and the sort of things you may not be able to do online.