Does any country have free college & open admissions?
Here is an overview of the situation in Germany, where there are no tuition fees for Bachelor and Master programs at public universities (though student union fees and public transport fees totaling 60-130 EUR per semester still apply).


For Bachelor degrees, the general requirements for admission to any German university are (1) proof of knowledge of the language of the study program (German or English) and (2) university entrance qualification (Hochschulzugangsberechtigung). 

"University entrance qualification" means proof that you are eligible to attend a university. In Germany, this is obtained by completing the Abitur examination. In the German education system, at the age of 10, students are already separated into different types of schools based on their performance, some of which have direct access to Abitur (and therefore to university), and others which have practically no access to university. About 50% of German students complete Abitur and have directly access to university. 

For international students, this is the corresponding high school leaving certificate. Depending on the country, your school leaving certificate might directly qualify you to study in a German university, or you might be required to do a two-semester preparatory course (Studienkolleg) before being allowed into German university . In this database (only in German), you can search for your country and school leaving certificate to see how it is in your case. 
For most Bachelor degrees, admission is unrestricted (Zulassungsfrei), which means that  any student that satisfies the above general requirements will be admitted, without going through a field-specific aptitude test. Switching programs after the first year is not difficult, but there is a somewhat strict requirement to pass the first-year courses in a limited amount of time, so the idea is to let students "test" their preferred program and switch to another one after the first year if they are not comfortable with it. 

Note that not all Bachelor programs have unrestricted admissions: in some fields such as Medicine there is a limited number of admissions and students will be chosen according to their grades in previous stages. This is called Numerus Clausus (NC).
On the other hand, most Master programs do have an aptitude test and students are admitted based on qualification in their field (in addition to the general requirement of having a Bachelor's degree). At this stage, students are supposed to already know what they want to study. I would expect the level of "strictness" of the aptitude test to be in some way related to supply and demand (disclaimer: personal opinion here), so the requirements do not necessarily have to be stricter than in US universities.  The offer of Programs taught in English (which are mostly only Master programs) is more limited than for German programs, so the aptitude test for English programs might also be stricter.

As pointed out in the comments, many Master programs are the result of the splitting of old 5-year programs (Diplom) into a 3-year Bachelor and a 2-year Master. Students doing the Bachelor program that apply to continue with the corresponding Master program at the same university usually have very easy admission into the Master program.


In summary, the requirements for entrance to German university in general are somewhat strict, but once you are in, there are in most cases few additional requirements for choosing a specific university or field.