I've heard Neil Peart talking about a "floating snare" section in his famous solo, in which there is no real sense of tempo or meter. This got me wondering if there existed any genres of music (whether in the distant past or today) in which the music is entirely "floating" with no meter / tempo. Does anyone know of any such genres?
Gregorian Chant is basically meterless. There is underlying rhythm among phrases but no long meter like in dance music or marches.
Perhaps unsurprisingly, John Cage has been there with "ASLAP":
The last note change occurred on October 5, 2013. The next change will not occur until 2020. The performance is scheduled to end on September 5, 2640.
https://en.wikipedia.org/wiki/Drone_music#Overview has links to a number of sub-genres of "drone music" ranging from Australian digeridoo and Scottish pibroch piping through to "experimental post-rock".
16th-century composers often wrote music without any regular meter, though it usually has an identifiable "tempo" or "beat" (as does plainsong). See collections like the Fitzwilliam Virginal Book (available free on IMSLP) - look for pieces with titles like "Fantasia", not the dances or sets of variations on popular tunes of the day.
There are many "modern classical music" examples written in a similar style during the last 100 years or so.
Most shakuhachi music is closer to having no meter than any other genre I know.
When having "abandoned strict tempo": A bene placito, senza misura, ad libitum, "in free time", et al.
Analysis of germane semantics on this one will find us splitting hairs, as it is a fine line separating "no meter" and modified tempo.
Music has traditionally three elements: melody, harmony, rhythm. You can imagine a jazz trio, sax(melody), bass (harmony), drums (rhythm), now, you can remove one or two. As long as they do sound (or silence) with their instruments, today is considered as music. You can even remove all three, today silence is considered also music (listen john cage's "3 minutes 44 seconds" on YouTube. Clasically, there were strict rules about how the three should behave, in order to consider it as music (european school, mainly). Today you can kick the door and present it as contemporary music. As long as it is intended to communicate feelings (that is art, in general), using sound, noise or silence, it can be accepted as music.
With classical Indian ragas, there is a first section, called Alap which is unmetered.
The best example of meterless music in my opinion is Electroacoustic Music. The core principles of the genre are that it has no discernible rhythm or melody. John Cage is one of earliest pioneers of this genre.
I studied Music Technology at university and one of our modules was on Electroacoustic Music. We had to produce a 6 minute song in this style, which I personally despised, yet somehow I produced one of the top pieces in my year.
Whilst this is a piece rather than a genre, I'd suggest the very opening of "Shine on you Crazy Diamond (part 1)" from Pink Floyd's "Wish You Were Here".