What do you mean? A country per se cannot be bought. The wealthiest man, Gates, has a net worth of around 67 billion USD (forbes). What could he buy?
Whatever it the country produces. The poorest country by GDP per capita is Burundi, with a total GDP per capita of around 267 USD, and a population of 10 million. That gives then a total GDP per year of 2.7 billion USD. Gates could easily afford that.
We don't have - to my knowledge - reliable wealth data on most developing countries, but the wealth/income ratio is typically around 4-6. Let's assume the maximum, 6. That would give the people of Burundi 13 billion USD, still affordable.
Let's pick a random country, switzerland. They have public assets worth 200 billion USD. Too much for Gates to afford. But for a smaller country, he could surely buy it out.
The answer is somewhat unsatisfactory. Why? Because the question is. In the end, before asking can xyz really happen, you should ask what is xyz and what do I mean by it?
What else is there?
I'm not aware of a country where the people are still counted as property, so you most likely can't buy them.
Perhaps you could buy all the land? There are many small islands (potentially for sale) that would grant you independence. Some people (mostly rich and eccentric) own them, which means I presume that they bought those countries at some point. So, it turns out,
You can buy a country
but you can't buy the people. !Viva la revoluçion!