If we enlarge a big rocket by a scale factor of two for all diameters and heights what will happen? All area will increase by a factor of four and all volumes and weights increase by a factor of eight. For eight times the mass we need eight times the thrust.
But if we take the same reliable rocket engines as before and don't build, test and debug larger engines, we need 8 times of engines as before. For a super size Saturn V we will need not 5 but forty engines. But we got only the double diameter of the rocket and 4 times the area and may mount only 20 engines instead of the needed 40 engines.
So we build new engines with double diameter and height and mount 5 such engines. Mass flow through the pipes, valves and nozzles will increase by a factor of 4, the same as the areas. The combustion chamber pressure will be the same. But using 5 engines with four times the thrust we can not take off, we need 10 engines with 4 times the thrust. Therefore we need new engines with double diameter and eight times the thrust. We need more powerful turbo pumps to get eight times the mass flow through pipes of double diameter.
If we want 8 times the thrust for an engine with the same combustion chamber pressure, we need to increase the engine diameter by a factor of 2.828 ( the square root of 8), the same scale factor for all pipes and valves diameters.
What about parallel staging as used by the Soyuz mentioned by uhoh?
A Soyuz booster was about 2.7 m diameter and 19.6 m high, it used four rocket engines. If we bundle 64 of those boosters in a square of 8 by 8, the edges of this square are 21.6 m, that is more than the height of 19.6 m. The total number of engines is 256. I don't believe that a bundle of 64 boosters could be mounted, transported to the launch pad, fueled and launched with success.
The next step would be a bundle of 16 by 16 boosters, that is 256 boosters and 1024 engines, 43.2 m wide(horizontal). Of course these bundle has excessive atmospheric drag and does not make sense.
So we can't win by using a huge number of engines with the same size, doubling all dimensions of an existing engine does not help. Using engines with 2.828 diameter requires a rocket of double height and diameter but 2.828 the diameter of the lower end to mount the engines.
The Saturn V was 110 m high and 10 m diameter. If we scale up the Saturn V by a factor of 2, we need 28.2 m diameter at the lower end. The ratio between height and maximal diameter was 11 at the original size and is now 7.77. What happens to this ratio if we double the size several times?
After two steps the ratio is 5.5, after 4 steps 2.75 and after 8 steps the ratio is only 0.6875. The rocket's diameter at the engines is now larger than the height of the rocket. This huge rocket would not only be ugly, the drag would be much too high. Acceleration to high speed would be impossible in the atmosphere.