What is the fastest beats per minute we can hear before it sounds like one continuous note?

Its is something called the Haas-effect. Lots of people here mention Hz, and while that is certainly related you are still able to distinguish the individual beats at a low frequency.

This is also known as the Precedence effect:

The "precedence effect" was described and named in 1949 by Wallach et al.[3] They showed that when two identical sounds are presented in close succession they will be heard as a single fused sound. In their experiments, fusion occurred when the lag between the two sounds was in the range 1 to 5 ms for clicks, and up to 40 ms for more complex sounds such as speech or piano music. When the lag was longer, the second sound was heard as an echo.

So the real answer is, depending on your metronome sound it will range from 1 ms (60000 BPM) to around 40 ms (1500 BPM) between each click where you can no longer distinguish each hit.

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