I answered this question on "is there a longest geodesic" by a kind of a joke, which I couldn't resist: the long line! It was a joke because I didn't know of a theory of such 'large' manifolds and I thought my intentions wpuld be clear - they weren't apparently.

But then, much to my surprise, whilst researching something else, I came upon a classification of such 'large' surfaces! Unfortunately I didn't make a note of it. However, whilst trying to find where and what this result exactly is, I came upon a paper by Rafael Dahmen on embeddings of the long line in weakly complete vector spaces. These are spaces like $R^I$ for arbitrary $$. The paper is here and titled: Smooth Embeddings of the Long Line and other Non-Paracompact Manifolds into Locally Convex Spaces.

Q. What I'm after is a paper or a book outlining the clasdifying such large surfaces.

I recall the source that I had looked at said it was summarising the findings of a certain book. Unfortunately, I don't remember the title of the book or which spurce it was - other than the source was online.