Is this the infinite dimensionality you refer to? I asked ChatGPT4:

In the context of dimensional analysis, what is "infinite dimensionality"? Keep your answer brief, only one or two short paragraphs.

In the context of dimensional analysis, the term "infinite dimensionality" typically doesn't refer to the physical dimensions (like length, time, and mass) that one might immediately think of. Instead, in mathematical and theoretical physics contexts, it often pertains to spaces, particularly function spaces, that have an infinite number of dimensions. For instance, the space of all possible functions of a particular type can be thought of as having infinite dimensionality, as each function can be uniquely described by an infinite series of coefficients.

One of the most commonly encountered infinite-dimensional spaces in physics is the Hilbert space, used in quantum mechanics. Here, the state of a quantum system can be represented by a vector in an infinite-dimensional space. While the idea of "infinite dimensionality" might sound abstract and counterintuitive, it provides the necessary mathematical framework to accurately describe and predict a variety of physical phenomena.

That looks off the mark to me - ChatGPT is jumping to the idea of R^∞ spaces and the like. I have no idea what the concept of a "meter^∞/second^∞" dimension would mean.

Well, this gets a bit heavy and is an area I have only modest familiarity with and not all that much practice.

That concept of infinite dimensional meters per infinite dimensional seconds doesn't work. With infinite dimensions that the whole idea of metrics breaks down. This gets used as a tactic: If you want to present some basic spatial concept without defining and then being bound to a specific metric then you might use infinite dimensional spaces for that since they are free of traditionally considered metrics.

This is rather exotic territory. The vast majority of phenomena that people find interesting can be characterized using eight or fewer dimensions.

I suspected so and I hope m0llusk will clarify. I found it frustrating to hear about an interesting missed case without enough context to google, unless you already know about infinite dimensionality in the first place.

"Dimensions" in dimensional analysis are not the same thing as dimensions in linear algebra! I'm not sure what a scalar quantity with an infinite dimension would be. (To reiterate, the dimensions of dimensional analysis tell you how quantities scale under adjustment of our scale of measurement, so having infinite dimension would mean scaling infinitely quickly…?)

That being said, I did try to connect infinite-dimensional problems with dimensional analysis in this post. Towards the end of my section on dimensions with complex powers, I applied dimensional analysis to the study of periodic signals and ended up arguing that (complex) Fourier basis elements can be thought of as dimensionful objects. I haven't seen dimensions used like this before, so I was hoping to earn the vote from fans of functional analysis ;)