The question is in the subject. Can music be arrhythmic(metrically) by the meaning of classical music theory?
The answer will be 'yes' for any definition of arrhythmic, as there are no rules about what music can and can't do - but it's probably still worth noting that it's not totally clear what 'arrhythmic' means without being more specific about the definition. Only a limited number of music styles are based on an entirely regular meter, with most styles allowing (or even assuming) a more flexible timing 'grid' on which events can fall. And even when we consider music with very little in the way of rhythmic repetition that would allow a meter to be discerned, the human brain is still likely to pick up rhythmic patterns from the timings of most possible clusters of events.
Some might say that a piece of music with a very low density of discernable events, such as an entirely steady drone, is 'arrhythmic', but others might require that for a piece of music to be 'arrhythmic' it should actively exhibit events that don't seem to adhere to a rhythm.
Yes. There are a variety of ways metrically arrhythmic musical events can occur.
Another example is Joshua Banks Mailman's "comprovisation" series, in which sensors are attached to the body, and musical meta-elements are controlled by body movements. (By meta-elements, I mean that movement doesn't directly affect the pitch, rather it affects the rate of change of pitch.) The technique is fully described in his article "Cybernetic Phenomenology of Music, Embodied Speculative Realism, and Aesthetics-Driven Techné for Spontaneous Audio-Visual Expression" (Perspectives of New Music vol. 54, no. 1, pp. 5–95 ).
The Wikipedia article on aleatoric music provides a wide variety of examples of music written in this way.
Music in "free time" is performed without a strict underlying pulse (meter); rather, the rhythmic timing is intuitive. The concept is discussed on SE MP&T: Free time in music.
The Wikipedia article also gives a wide variety of examples.
The core idea behind a cadenza is that it be improvisational; thus, the meter may not be strict, and specific rhythms may or may not be discernible.
For example, see the following SE MP&T Q&A: Why aren't there barlines in the piano solo parts in Beethoven's Piano Concerto No. 5?
Rubato is the stretching and compressing of musical time for expressive purpose. Although it maintains the rhythm, it can obscure the meter.
Can music be arrhythmic(metrically) by the meaning of classical music theory?
I don't think you need to bring in "classical" theory. The essential meaning of rhythm is periodicity, something that repeats. You just need to have some music with no repeating element.
That could get a bit tricky philosophically, because you need to ask what elements repeat and what time scale are you considering. Even the theory of meter can get vague about what specifically creates the accent patterns of meter. In a certain way it is just presupposed to exist conceptually.
A simple beat is rhythmic, it repeats. Tempo changes don't necessarily mean arrhythmic. An accelerando and rubato change the tempo, but you wouldn't describe those as arrhythmic.
A drone seems an obvious possibility for arrhythmia, but even things like didgeridoo, bowed string, circular breathing have patterns. Eventually you breath in, the bow changes direction.
The human mind tries to find patterns (rhythms) and it seems there is significant flexibility about what can be identified as a pattern. I think you need to get into the avant-garde to have music that could truly be considered arrhythmic. You could go the John Cage route and call complete silence your music. You could turn on some sine wave generator and have drone a steady pitch and call it music. White noise: music. Etc, etc.
One final thought. Even in computer programming people will point out your can't have truly random numbers.
On a superficial level it seems easy to be random, but on a deeper philosophical level it's difficult.
European classical music theory could perhaps withstand the removal of the concept of rhythm, but it definitely depends on the concept of sequence, of events happening in a particular order. Consider: most aspects of the analysis of a chorale harmonization or of a melody's structure would be unchanged if the durations of all the notes were changed to random values. Indeed, much medieval music, notably liturgical chant, is preserved without any rhythmic information. On the other hand, if you change the order of the notes, the pieces would quickly lose their identity.
Despite this, European classical music (indeed, European music generally) has been fundamentally rhythmic since techniques for recording rhythm were invented, and probably before. Every other musical tradition with which I am familiar is also rhythmic to one degree or another. (There are several such traditions, though I have only passing familiarity with them, including music of Ghana, Karnataka, and Java, of indigenous peoples of the Americas, and to a lesser extent Chinese opera.)
It also bears mention that rhythm and meter are not synonymous. Much polyphonic music from the late middle ages and the early renaissance is not metrical in the modern sense, but it is clearly rhythmic.
An important division of extemporaneous music (manodharma sangita) in the Indian classical tradition is rāga ālāpana. In rāga ālāpana, the musician develops a rāga in a step-by-step fashion while obeying the structure and characteristics of the rāga, and this is done without any strict rhythm. So, this could be another example of arrhythmic music, to add to the other nice answers to this question.
Drone music can fit that pattern. A single note or chord in this type of music can ring for so long before another one is struck that all sense of rhythmic connection between the two "events" is lost. This is also the case for ambient music in general (of which Drone is technically a subgenre). Check out Sunn O))) meets Nurse with Wound - The Iron Soul of Nothing for an example.
Noise music can also be arrhythmic by definition of it being noise, although at the same time it's by definition not music, so I'm not sure it fits your criteria.
As others have pointed out, the rhythmic feature is related to pattern formation in the music. As cited in the examples of trying to form "chaotic" "enough" music, most forms of attempt with human guidance would end up forming patterns, be it from the instrument played, or the human player. Mathematically speaking (and as an amateur mathematician), it would be actually harder to create "perfect" chaos than to create "imperfect" chaos, which would actually not be chaos at all. This makes the answer harder to give because of lack of examples. The theme of 'absence of regularity' pertains to chaos theory and is advanced, it seems. For example, I imagine it would be an interesting mathematical question whether and how difficult it would be to create music without any trace of regularity across all its axes. It could be impossible, and then you could say it features some kind of rhythm somewhere. But I don't know how much classical music theory incorporates of mathematics. But it would be as hard to prove a piece of music as anything else that it is chaotic.
I hesitate to pretend to speak on this, since I know nearly nothing about it, but I am dimly aware that there are Africa-continent (and probably other) musical styles that involve mostly drums and percussion instruments, but/and at significantly different repetition rates, so that the whole thing only repeats itself after a considerable interval of time. In particular, listening to just some moments of the pieces might seem to be a cacophony. I hope someone here knows more about this.
(Question, then: does lack of small-scale pattern discourage us from believing in a large-scale pattern, by instinct?)