Which airplanes require two (or more) pilot/operators?
The question boils down to, 


  "Are there any critical flight controls that are out of reach of the
  Pilot In Command?"


Some older aircraft that required a Flight Engineer in the third seat (not the copilot) and controls at that station that were not available to the pilot or copilot. Some older variants of the 747 (pre 400 series, not sure if any are still in service in their original config) had a required 3 man crew. The Concorde (when it flew) had its fuel system controls at the FE station and those had to manned in flight as the fuel had to be moved around during different stages. You may be more likely to find these older airframes/configurations in cargo service these days. For these planes the legal requirement is often 3 people (not 2) but from a pure "get it off the ground, fly it, land it" case you need at least 2 people to operate it.  

The FE station looks something like this in most planes (747 in this pic) 

(source)

The Concorde had a pretty daunting panel as well 


All that being said you would need at least a Pilot and FE to physically fly the plane since the pilot cant reach all the controls.

Modern avionics controls (glass cockpits mainly) have allowed more information to be presented directly to the pilot in a more condensed manner and allows displays to have multiple functions unlike gauges which tend to display only a single datum.