What is the maximum amount of noise an aircraft is allowed to generate?
If it can be regulated it shall be regulated -- the relevant section of law is FAR 36 - NOISE STANDARDS: AIRCRAFT TYPE AND AIRWORTHINESS CERTIFICATION, with a number of advisory circulars as well.

These requirements are aimed mainly at Transport Category aircraft, and the absolute noise level limits are rather high (certainly nothing you'd want to be standing near without hearing protection).

What's probably of more interest to most people are the FAA's guidelines on Airport Noise, and specifically FAA Order 1050-1E, which deals with the noise you are actually exposed to on the ground (rather than what is generated by the plane's engines).
Order 1050-1E goes into great detail on what is considered "acceptable" noise levels for various types of land use around airports.  Airport construction and/or expansion projects are evaluated against these criteria using computational models to ensure that noise levels will not exceed the FAA's guidelines.