How far can a 777 fly with just one engine at 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.