We all know that, at least theoretically, there are several possible models of computation, varying in structure.
Strictly speaking, there are several (not just one) models of computation that exist or have been implemented in real life. Some machines (for example, amplifier circuits), though inexact, can be considered analog computers, but that's not what we commonly think of as 'computers', is it?
The machines that we usually think of as 'computers' are implementations of a computation model with discrete parts (discrete memory slots, discrete instructions and clock cycles, etc.) I've always wondered if there's some form of data that can't be represented/stored inside these discrete structures.
I've brought up this question in many websites, but I usually only find people who approach this issue with an incomplete picture of the problem, and without the required logical rigor.
I've been told that nothing with a continuous structure can be stored inside a computer, but from what I've gathered, this is not always true. Take the example of a circle (a geometrical shape). It's true that it has an infinite number of points in its circumference, but I can't say that they represent an infinite amount of information, since they can be derived from a very small number of parameters (center and radius) and a bunch of simple laws (algebra and symbolic math). My intuition tells me that this is the case for any geometrical shape, unless it has an infinite number of parameters, in which case I don't really know what to think.
Is my intuition accurate? Is this the case for any structure with a finite number of parameters? Are Von-Neumann/Harvard/... computers unfit to store certain data structure(s)? Has this been mathematically proven? Is the proof constructive? Are there any data structures that can't be stored in any known model of computation (or, for that matter, any model of computation)?
Note: I'm asking all of this from intuition. Even though I write code for a living, and harbor a great interest in Computer Science (and have a little knowledge of it through self-study), I am not a CS or Math graduate. From my limited, newbie perspective, I would think that this question belongs to the Theoretical Computer Science Stack Exchange, but I genuinely don't know if it's too basic, so I'll just play it safe for now.