Discussed a number of times in the past, but the good thing with this list is you can have the details. If one look into them, one notices that many countries accepting foreigners in their armed forces have some form of conditions: having dual citizenship or legal resident status, being a member of the EU for some European armies, being from certain countries (ex-Warsaw Pact for Russia, Iceland for Norway, Latin America Spanish speaking countries for Spain, Commonwealth for the UK, etc.), being Jew or having some form of Jewish descent for Israel, etc.
The Legion, has some unique or rare specificities :
• It's the only military unit accepting recruits from any country in the world, without any pre-requisite,
• It's also one of the very few forming a specific branch (along with the Gurkhas, but I don't see any other example). Otherwise, foreigners, in most cases, are simply embedded in regular military units, besides locals.
• And last but not least, it's the most famous worldwide.