If you come to the USA without a visa, (under the visa waiver program that currently exists for citizens of many countries), you can stay only 90 days maximum. The length of stay cannot be extended. If you leave the country for a short visit to some other country like Canada, Mexico or the Carribean Islands, you can reenter the USA, and stay only the days remaining on your original 90 days. Technically, if you return to your home country for even a short time, you should be able to return to the USA for another full 90 days.
If you want to visit the USA for more than 90 days, you should apply for a visa. The US embassy in your home country can issue you a visa for a 90-day visit to the USA. When you are in the USA, you can apply to extend that visa for a longer stay. USCIS Form I-539 (Application to Extend/Change Nonimmigrant Status) As far as I know, there is no limit on the length of the extension. When you file your application for permission to extend your stay with the US Citizenship and Immigration Service (USCIS), the bureaucrats will determine how long you can stay. Those bureaucrats have the power to grant nearly unlimited lengths of stay. In the case of students, they often grant visa extensions for the duration of status, which means the student can stay until they complete their studies.
When you file your application for extension, you should try to give them very compelling reasons for why you wish to stay in the USA.
Here is a link to the USCIS web page describing visa extensions.