If a 777 pilot wanted to fly further than with both engines running, how many more miles could the plane go with just one engine running, at high altitude?
Flying with only one of two engines will not increase range, but reduce it. A gas turbine consumes fuel in proportion to the mass of air moving through it, so with increasing altitude consumption decreases proportional with air density. Thrust goes down with density as well, so thrust-specific fuel consumption is roughly constant (actually, the engine becomes more efficient with altitude due to the lower air temperature, see Carnot cycle for more on that).
Generally, a jet transport wants to fly as high as possible for maximum range. Its cruise altitude is limited mainly by the thrust available from all engines. If the pilot shuts one down, the aircraft needs to fly much lower until thrust has increased sufficiently to overcome drag. Since friction drag is also proportional to air density (and the square of flight speed), the aircraft has to fly much slower at that lower altitude, which cuts range since the fuel consumption of the single engine is now roughly what both consumed at altitude, but now the aircraft covers less distance in the same time. In addition, the pilot has to compensate asymmetric thrust with rudder deflection, which again increases drag.
Short answer: The aircraft will not go MORE miles, but FEWER.
It depends on the airliner design. If one engine is inside the fuselage tail section and can close side intake scoops and streamline engine bypass cowling, then range will increase as drag is a lot lower with one engine out of the air steam, and the remaining outside engine will have a higher engine rpm, hence higher efficiency using less fuel for the needed thrust. Chances are the needed thrust will be available as engines spool down quite a lot at cruise (oversized to take off). However, the higher thrust will decrease the life of the single cruise engine. Of course, the single cruise engine could be sized to take advantage of this condition to maximize fuel efficiency at normal wear.