What are the advantages of more than 4 propeller blades?
You are right, more blades are bad for efficiency (follow the link for the definition). Ideally, a propeller should have only one blade. Every additional blade will cause disturbances which interfere with the flow on the other blades.

When engine power increases, the propeller disc area should also grow, but this growth is limited by the resulting speed of the blade tips. Once the flow speed there becomes supersonic, the drag at this section of the blade increases without a corresponding increase in thrust. To avoid that, the next best option is to increase the solidity ratio of the propeller, called also the activity ratio. This is done by either increasing blade chord or the number of blades.

Make no mistake, this is bad for efficiency. But if there is enough power available, adding more blades is the best way to transform that engine power into thrust.

Take the Supermarine Spitfire:


The prototype, powered by the 1,030 hp Rolls-Royce PV-12, had a two-bladed propeller
From the Mk II, a three-bladed propeller was fitted to accommodate the increasing power of the RR Merlin (1,470 hp for the Spitfire Mk V).
With the Mk IX, a four-bladed propeller was needed for the 1,575 hp of the supercharged Merlin 61.
From the Mk XII, the more powerful Rolls-Royce Griffon made a five-bladed propeller necessary. Engine power was raised from 1,735 hp to 2,300 hp for the last variant, the Mk XXIV.




Rolls-Royce Griffon-powered Supermarine Spitfire (picture source)

A lower prop speed allows to increase its diameter, but while tip speed will drop by less than the reduction in prop speed (after all, flight speed should not change), the available thrust from this propeller will drop by the square of the speed reduction, since thrust is proportional to the dynamic pressure on the blades. And thrust you get only from the circumferential fraction of the local speed at the blade; flight speed does not count here and does not help to mitigate the reduction. Consequence: You cannot make the propeller bigger and spin it more slowly for a given power.

An extreme example for a propeller with a high activity ratio is the Aerosila SV-27 contra-rotating propeller of the D-27 propfan engine powering the Antonov An-70:



SV-27 propellers on the An-70 (picture source). Eight blades in the forward disc and six in the rear, running at only 1200 RPM. In order to reduce Mach effects, all blades have a swept tip and deep chord.

This can only be topped by something like an ungeared turbofan with a ducted propeller. A turbofan, in other words.