There are no explicitly regulated MTOW limits for a private pilot with a valid medical but extra certifications/training may be required to fly heavier aircraft.
The often cited number is 12,500 but this is somewhat incorrect depending on how you look at it. You will need a Type Rating to fly anything over 12,500 but strictly speaking those can be added to a private pilots license.
If you are a PPL flying under Basic Med you are limited to 6000 LBS MTOW
14 CFR § 61.113 - Private pilot privileges and limitations: Pilot in command.
(i) A private pilot may act as pilot in command of an aircraft without
holding a medical certificate issued under part 67 of this chapter
provided the pilot holds a valid U.S. driver's license, meets the
requirements of § 61.23(c)(3), and complies with this section and all
of the following conditions and limitations:
(1) The aircraft is authorized to carry not more than 6 occupants, has
a maximum takeoff weight of not more than 6,000 pounds, and is
operated with no more than five passengers on board; and
Just a note on practicality, even if you don't get into the 12,500 range you are likely to need high performance and complex ratings to fly anything much bigger than a 172. While there are some big fixed gear aircraft out there most of your common 6 seat planes (Saratoga, Bonanza, etc) are retracts.