Lev asks, "I am a citizen by birth of a European country but because my mother is Asian, I also automatically became a citizen of that country. Now, I am ready to file for naturalization in the United States, and am wondering what will happen to my two other nationalities. Will the United States force me to renounce these nationalities? Is there an upper limit to number of nationalities that I can have after becoming a US citizen? Will the US Government contact authorities in these country and let them know about my naturalization in the US?"
Assuming that you meet all the requirements of US citizenship and are approved, the United States Government does not care what other nationalities you have. While the country does not have an official policy, it does recommend that aliens who become US citizens do not pursue citizenship in other countries. What will happen to your citizenship in other countries will be determined by them. For instance, Japan will let a Japanese citizen retain her citizenship. However, Indian law says that you lose your Indian citizenship as soon as you take the oath of citizenship of another country, though, if you plan to travel to India after naturalizing, the process requires you to officially renounce your citizenship (no paperwork is needed if you have no plans to ever travel to India, though, it is always a good idea to let your native country know that you are no longer their citizen).
The main point to remember is that while having multiple passports has some privileges (visa free entry in some cases or be able to own property, etc.), all rights come with responsibilities. You may have to pay taxes in that country or be prepared to join the military during a war or be bound by their laws when present in that country (which may be harsher for their citizens, for example, many countries insist that their citizens receive exit visas to leave the country). While many countries will never let you renounce their citizenship and insist that you even use their passport to visit them, it is best to keep things simple if possible by having just one nationality.