Are there any countries who will allow anyone to immigrate into their country and eventually become citizens?
As TimSparrow's comment noted, there are countries which allow people to buy citizenship. Actually, nearly all countries do including USA and Europe, but some are more choosy about who they accept/vet/reject than others - such as Mexico or Canada. On the other hand others are less choosy:
Dominica was selling their passports to any comers at all.
This was discussed in more detail here: Why countries selling passports is bad?
Ecuador: To immigrate, all you need to be able to show is a guaranteed monthly income of USD $800 for yourself, and an extra USD $100 for each of your dependents.
Argentina: To settle here, all you need to show is a guaranteed monthly income of USD $850.
Uruguay: they also require a guaranteed minimum income to immigrate (couldn't find out how much yet). But it's not trivial to become a citizen later, though nowhere near difficult:
Applying for Uruguay citizenship requires that you show proof that the country is your center of life. This proof can border on the arcane, as judges and immigration officials have requested library cards, receipts from doctor appointments, or proof of country club memberships. (source: http://nomadcapitalist.com/2015/07/22/how-to-get-uruguay-citizenship/)
However, there are no countries that have a 100% free immigration policy - they all restrict who can immigrate either by required guaranteed income (many Latin American countries); or skillset (Canada, Mexico etc...), or a hodgepodge of policies (US, EU).
P.S. you can freely immigrate from one EU country to another, but that doesn't apply to non-refugee immigration from outside EU. Actually, not that freely as comments suggested - you still need a source of financial support.
All citizens and all companies of every nation under the treaty are allowed to become residents and to have access to Svalbard including the right to fish, hunt or undertake any kind of maritime, industrial, mining or trade activity. The residents of Svalbard must follow Norwegian law though Norwegian authority cannot discriminate against or favor any residents of any given nationality.
Of course there's a catch to this seemingly perfect getaway destination. To quote the excellent answer by @jpatokal on Travel.SE:
- Located at 78°N, Svalbard's main town Longyearbyen (pop. 2600) is by far the world's most northernmost civilian settlement, and rather resembles Mordor after it has frozen over: lots of black rock, virtually no plants larger than lichen. This means an average February day is −21°C, a balmy summer day in August is 3°C, and oh, there's no sunlight whatsoever during the polar night between late October and mid-February. Also, you need to carry a rifle if traveling anywhere outside the town due to the large and hungry polar bear population.
- The only practical way to get to Svalbard is Norway, which can and does enforce its own visa rules. That said, once there Norway and Schengen rules (eg. the 90-in-180 limit) do not apply, all you need to demonstrate is that you have the means to support yourself.
- Essentially all private accommodation on Svalbard is owned by companies in Svalbard, which means finding a place to rent is tough to impossible. Now you could just stay in a hotel, but...
- If you thought Norway was expensive, imagine Norwegian prices with a hefty bonus for shipping everything through some of the roughest sea on Earth. You won't get much change back from US$20 if you buy a kebab and a Coke.
Also note that Svalbard currently treats citizens of non-signatory countries the same as citizens of those that have signed, so it covers 100% of the world's population. It used to be possible to become a Norwegian citizen after spending seven years on the island, but this "loophole" is currently closed.