It can happen when the groupings of note change per measure and don't stress any exact metric pattern.
While not common, it can happen and the most famous example of this is in the 3rd movement of Quartet for the End of Time .
As you can see the tempo is defined, but the meter is not.
The eighth note gets the tempo markings, but it's not technically the beat because there is none.
Listening to it should give you an idea how it flows.
The most interesting thing to me is how the bar line is kept not because of the metric hierarchy, but to show accents in place of it.
music.stackexchange.com/users/7222/Dom
Indeed, some songs' time signatures cannot be neatly categorized into 3/4, 4/4, 2/4, 9/8, 5/4, etc.
Songs in free time aren't notated with any time signatures at all.
Satie was pretty fond of writing such pieces.
Songs in mixed meter are notated with time signatures, but they change time signatures as necessary (and sometimes pretty often).
Perhaps covertly the most famous mixed-meter song is "The 12 Days of Christmas".
Other mixed-meter songs that stick out in my mind include Stravinsky's "The Rite of Spring" and Dream Theater's "Ytse Jam".
music.stackexchange.com/users/37354/Dekkadeci
Would you count music without even tempo?
Lots of Penderecki has just minutes and seconds of elapsed time marked off.
There's no meter or tempo.
And there are lots of vertical dotted lines to show where the different changes in the music line up between instruments.
Also the theme to
The Terminator is based on a loop that doesn't really fit in a time signature or meter.
music.stackexchange.com/users/6070/Todd Wilcox
One genre where a time signature is completely alien is plainchant:
This even applies where "chant" is written out on five lines: English has time-based stresses, so the notes are compressed in some cases (like "God" in "Let God arise") so that the stresses in the words come as they would be spoken.
The stressed syllable "rise" is as long as the three notes on "God a-", in order that the next stressed syllable, "let" falls at the right time.
But that would be extremely complex to notate in the normal way, so a different convention is followed.
Images: Sanctus from Mass XVIII, my origination;
Introit for Sunday 17 by Robert Rice in his collection St Michael Antiphons .
music.stackexchange.com/users/2731/Andrew Leach
To add to Andrew Leach's answer, another historical example of music without time signatures is the French prélude non-mesuré for harpsichord or lute, which was in vogue during the mid=to-late 17th, early 18th century.
Composers of such pieces didn't even notate rhythms, leaving those up to the performer.
(Although of course certain implicit stylistic conventions would have been assumed known to any competent harpsichordist or lutenist.)
An example of an unmeasured prelude with sound and notation.
music.stackexchange.com/users/24130/Kim Fierens
My understanding is that a time signature like 4/4 time is known because certain beats are emphasized, for example:  ONE , two, three, four. But let's say someone created a song with tempo alone (ie just used a metronome) and didn't emphasize any beats. Then can a song's time signature be simply unknown and cannot be categorized into 3/4, 4/4, 2/4 etc?
Are there situations where a time signature is unknown?
music.stackexchange.com
<notation><rhythm><time-signatures><tempo><meter>
music.stackexchange.com/questions/73269
music.stackexchange.com/users/None/
Songs in free time aren't notated with any time signatures at all.  Songs in mixed meter are notated with time signatures, but they change time signatures as necessary (and sometimes pretty often). Indeed, some songs' time signatures cannot be neatly categorized into 3/4, 4/4, 2/4, 9/8, 5/4, etc.   Songs in free time aren't notated with any time signatures at all.
Songs in free time aren't notated with any time signatures at all. Some songs' time signatures cannot be neatly categorized into 3/4, 4/4 or 2/4.