There is multiple ways of looking at this. If we see lowest note as lowest possible frequency — well the lowest frequency is 0Hz. So what note is this? Well since 0 is a multiple of any frequency it is every note there is at octave -Infinity. Of course, if we are talking about non-constant vibration there is no lowest frequency. We can go as close to 0 as is possible.
Now, one different way of looking at this would be to consider lowest note regularly produced by instruments. Issue with this is that we can very well just detune a string instrument to produce quite low frequencies. So really this is not well defined. Also some percussive instruments produce fundamentals that are very low, but not really audible.
Now, if we consider standard orchestra and piano the lowest note is usually the subcontra A (A0 at 27.5Hz), reached by instruments such as (some models) contrabassoon and bass Tuba (although this one can in fact go quite a bit lower in theory).
Then there are of course organs with 32' stops, which means the low C gives a frequency of 16.24Hz (although this should not be seen as an individual note, but as a component in a organ mixture). Also there are certain grand pianos such as the Bösendorfer imperial that go down the whole subcontra octave, giving the same 16.24Hz.
Then there are rare experimental instruments, like the subcontrabass clarinet and subcontrabassoon, going down to subsubcontra Bb (Bb-1, 14.57Hz). And of course there are two organs in this world featuring an 64' stop, which means you get a subsubcontra C (C-1) at 8.14Hz.
A different way to look at it would be to consider the lowest frequencies audible to human hearing, which is generally considered to be 20Hz with most people. This is not really true, but the lower you go the more sound pressure you need for the sound to be audible, and as your hearing gets worse this required level skyrockets. So this 20Hz (something between a subcontra e and eb) is not a point where you stop hearing a note — it just disappears as you go lower. This means that while 20Hz might be audible at still normal sound pressure, it will be very soft.
The problem with this definition is that human hearing does not actually need to hear the fundamental to recognize the note. The brain is good as completing the harmonic spectrum. Indeed, most speakers go down at most to some 50-80Hz (so something between a G1 and an E2). So the lowest pitch on many speakers is in fact the low string of the guitar. Still you’ve got notes below this played from said speakers. The low contra C in the double basses in Schubert’s h-minor symphony does not disappear, just because it is 32.48Hz, and you get electric bass in recordings all the time, even 5-string variants going all the way down to subcontra B.
So taking the limit for the fundamental as lowest note does not seem reasonable. If we demand to at least have the first overtone the lowest note would then be at about 10Hz, but even then, the brain could maybe do without.
Of course, going this low means that the brain is not really able to determine the pitch anymore, so these very low registers all only make sense for doubling things in the octave below.
There is yet some mathematical limit to pitch: Now, for an oscillation to have a base frequency f it needs to go on for at least one whole oscillation (although realistically you’d always want more than just one oscillation). This means you need to play it for at least 1/f seconds. Now, if we find a limit for how short a note should be playable then this gives a lower bound for frequency: Say, no instrument will be able to play a note that lasts 1000 years = 31557600000 seconds, so all notes must be higher than 1/31557600000Hz, which would be about a C♯-39. But then of course, this is a rather generous limit. We might argue that any musical note should at least allow some amount of rhythm on it, say, at least quarter notes at a reasonable tempo of say quarter = 60. Then the lowest possible note would be 1Hz, or a C-4. Of course this kind of depends on what the time limit is. If for example we should be able to 16ths at 60 qpm then lowest note possible would be 4Hz or a C-2.
In general: If your first define a duration d sec. such that any note should be possible to sound that short then the limit frequency would be 1/d Hz.