In addition to a regular 10-year passport, US citizens can request a limited validity (4-year) second passport.
Being in possession of more than one US Passport must be authorized by the US Department of State. The most popular reasons for having two passports are:
Being a frequent traveler and having to get foreign visas and travel, often at the same time.
Or, your primary passport may contain a foreign visa or entry stamp that would prevent you from entering a third country. For example an Israeli Visa or stamp in your primary passport would prevent you from getting a visa for Saudi Arabia
While it is highly unusual for the average US citizen to have 3 passports, there are instances we know of where this may be possible:
US government employees are issued an "official" passport when traveling on government business, and U.S. diplomats are issued a black "diplomatic" passport. Thus it is possible to have three valid US passports. For example, someone with two normal passports (a 10 year primary and a 4 year limited validity) for traveling on personal NON GOVERNMENT BUSINESS and one diplomatic or official passport for government business.
The US Dept of State highly recommends that even if you do have more than one valid passport, that you travel with only one whenever possible.