Recently, I have been considering incompleteness in many different contexts - I'm sure many would agree that incompleteness evidently arises in contexts where certain structures gain sufficient complexity that some well-known mathematical machinery, such as axiom-schemata or Turing machines, can no longer describe or solve every problem.
It led to consider the vague notion of whether there could be spaces (preferably topological, since they are nice to work with!) with the property that no finitary and consistent coordinate system can be made that can act as a notation for every point. In other words, any consistent coordinate system will always "miss" certain points that, nevertheless, exist. I presume here that a "coordinate system" would consist of some alphabet of characters and rules for string formation out of this alphabet, along with some interpretation function that induces an injection between the set of well-formed strings and points in the space (I also presume that such a system would identify a set of one or more "base points" with which all other points would (hopefully) be able to be described relative to them). I would call a space that cannot be described by such coordinate systems supergeometric spaces.
To be clear, I do not consider the space of polynomial functions to be supergeometric, since a polynomial can still be written down finitely. I have heard, however, that a phenomenon not unlike what I describe above appears to hold with the class of Ordinals and that there is a proof of the fact that all finitary notation systems for ordinals break before reaching the Church-Kleene ordinal. So, the Ordinals are perhaps supergeometric (since one can consider it a space when inducing the order topology) - are there other well-known spaces that are? Can we construct any?
It has been pointed out to me that, in fact, all spaces with an uncountable number of points have this property, in that if I have a finite size alphabet and then take the set of all finite-length strings that can be formed from that alphabet (Kleene stars, anyone?) then we still have a countable set - thus, we would never even have enough strings as points!
It was suggested in the comments that infinite-dimensional spaces are the answer to this, however I believe these to be a red-herring. We could have the infinite dimensional space of power series, but then limit ourselves to the (still-infinite-dimensional) subset of polynomials. It is clearly true that we can write down polynomials finitely, since the point (0,1,0,0,0,0,....) can be written as $x^2$ (here, we are essentially cutting off the trailing infinite sequence of zeros that is guaranteed to exist due to it being a polynomial). Similarly, as pointed out in the comments, the real-line cannot be described with this method and it is 1-dimensional! Thus, infinite-dimensionality is neither sufficient, nor necessary for supergeometricity.