What is the maximum legal flight time for single pilots under Part 91/135 operations?
Short answer: no limit for private Part 91 operations, otherwise 8 hours.

The only Part 91 regulations I can see on this are in Subpart K - Fractional Ownership Operations. There's too much information to quote here, but see 91.1057, 1059, 1061 and 1062. For a single-pilot operation the basic limit is 8 hours, per 1059(b):


  (b) Except as provided in paragraph (c) of this section, during any 24
  consecutive hours the total flight time of the assigned flight, when
  added to any commercial flying by that flight crewmember, may not
  exceed—
  
  (1) 8 hours for a flight crew consisting of one pilot;


There's no restriction for typical private GA pilots, apart from 91.13 on careless or reckless operation. Flying while very tired could easily be seen as reckless if it affects the safety of a flight.

As for Part 135, the regulations are in Subpart F—Crewmember Flight Time and Duty Period Limitations and Rest Requirements. Again, there's too much to quote but the basic limit for a single pilot is 8 hours, per 135.267(b):


  (b) Except as provided in paragraph (c) of this section, during any 24
  consecutive hours the total flight time of the assigned flight when
  added to any other commercial flying by that flight crewmember may not
  exceed—
  
  (1) 8 hours for a flight crew consisting of one pilot;


Your second question about when an additional pilot is required is a bit unclear. A second pilot is usually required because of the type of aircraft or operation, not the length of a flight. But I know very little about Part 135 so I may be wrong; in any case, I think it would probably be better as a separate question.