Thailand, Hong Kong, Singapore, Vietnam and Cambodia.
Have you ever explored your ancester's old digs?
Maybe that's why you're in Ierland now.

It is at least two of you. For three months and at least two people I think 10k EUR is not a lot. It's certainly not little money, but not a lot.
So you should stick to South America, Africa, or SEA.
Personally I'd do a self-drive safari through Namibia, Botswana, Zambia, Mozambique, Malawi, Tanzania, Kenya. (As far as you get anyway. All of them will likely be way too much)

new zealand!!! the summer is nice and warm, north hotter then south. so many different landscapes and so peaceful and safe. i am living here in nz and its one of the best countrys to travel. rent or buy a campervan and enjoy your freedom :)
Choose the Philippines... it's a World of Wonders...7, 101 emerald isles and one of the warmest people hearabouts...

Deffinitedly you can come to South America, wonderful spring time is comming so I suggest to visit Agentina, the country itself it´s so extensive that will take you the whole 3 months!! Other option, you can combine different countries of South America, impressive landscapes, culture diversity and within the mentioned budget, Highlights of South America are Peru (culture and impressive landscapes), Argentina (great landscpaes variety, glacier, falls, lakes, highest mountain in South America, cosmopolitan Buenos Aires city, amazing cuisine with great wines and BBQ - asados) and of course culture mix mix of European influences, after Argentina end trip at Brazil relaxing at a wonderful beach. Good luck planning the trip!

Go to US and buy a National Parks passport at the first National Park you visit. http://www.nps.gov/ Then get cancellations stamps in your passport whenever you visit a National Park. http://passport.eparks.com/store/product/22515/%2APassport-To-Your-National-Parks%2A/ Try to get a cancellation from every region if not from every state. http://nostalgia.esmartkid.com/passportstamps.html.
what about INDIA ? its one hell of a beautiful and warm place.... there is a lot of diversity in terms of Landscape, People, Cultures, food and places :) I assure you will have one of the most rejuvenating experiences of your life.......
I totally will deny that India is one of the cheapest places in the world.... But i can say that it has a lot of options in terms of budget is concerned. It has options from 1000 dollars a night for a hotel until minimum 5 euros a nite (but believe me the 5 euros a night is not that nicer place to live in)
If you are wanting to go to one of the cheapest places in the world then you could choose from a list of under developed countries :))))
But i would still suggest you India for a taste of the Royal Princely experiences of the Palaces, warmth of the Cultural traditions, Fast and furious lives in Mumbai, Delhi, Calm Backwaters of Kerala (rejuvenating Ayurveda Spa Experiences), temples, palaces, Colonial architecture of Kolkatta, Goa, Mumbai, Himalayan Mountains and Valleys, the beautiful sea and the islands of Andaman and Nicobar, Lakshwadeep.......... and many many such experiences as too many to pen down here.....
:))))