When a planes maximum range is specified, is fuel [capacity] the only constraint or what else is taken into account?
Would it always or ever be the case that a greater fuel capacity will increase range?
Interesting long-flight story of 64 days in the air, but it doesn't get too technical: https://jalopnik.com/how-a-slot-machine-mechanic-flew-150-000-miles-over-64-5971433
As also stated in this answer:
Range is a function of MTOW and empty weight, according to the Breguet Equation for a jet: $$R=\frac{V}{c_T} \cdot \frac{L}{D} \cdot ln\frac{W_i}{W_i-W_f}$$
with ${c_T}$ = specific fuel consumption, $W_i$ = initial weight, $W_f$ = fuel weight, $L$ = lift, $D$ = drag.
What increases range if other factors remain constant:
The weight ratio can be because of fuel weight used during the flight, but of course chucking ballast overboard helps as well.
The airplane's published range will be based on distance that can be flown in still air (excluding winds in other words) with full fuel, departing at max gross weight, when flown at some optimized cruise speed or mach#, at a specified flight level, with normal IFR fuel reserves (fuel to go from A to B, fly an approach, do a missed approach, proceed to alternate, fly an approach there, and land, with, for a jet, 30 minutes remaining after all that), in standard ISA atmospheric conditions.
Somewhere in the fine print in the airplane's specs will be the Mach# and altitude that the maximum range is based on.
Most airplanes can't be flown with full fuel AND a full cabin without being overweight, so you have to horse trade. A corporate jet with fuel for 5 hours endurance may only be able to carry 3/4 of the passenger complement to get max range, or conversely be limited to 4 hours if the cabin is full. Such a jet may have an option for a long range fuel tank add-on, increasing endurance to 6 or 7 hours, and if that tank is also full, the passenger load may be cut down to just a few passengers.
Speed, fuel capacity, and burn rate are the only three factors that determine range. Optimizing each of these variables will lead to the most efficient results. Generally speaking greater fuel capacity will increase range, which is why auxiliary tanks are sometimes used.
There are a great many design factors that influence each of these three variables, but that is beyond the scope of the question. Additionally, environmental factors such as wind will affect actual performance.
Would it always or ever be the case that a greater fuel capacity will increase range?
The problem for a flight without external assistance is you have to carry the fuel (and any other consumables but fuel is by far the most significant consumable). You can add more tanks (and ferry pilots do), but ultimately you reach a point where your takeoff weight is at the aircrafts limits.
You could make the whole aircraft bigger, but then it needs more fuel, so you haven't really gained much.
The 64 day flight you linked was not unassisted. They received fuel and supplies from a truck on the ground during the flight. It also sounds like it was rather unsafe, with the article reporting that one of the pilots dozed off at the controls due to inadequate sleeping arrangements.
