My intuition tells me that the only connected 1-dimensional topological manifolds are the real line $\mathbb{R}$ and the circle $S^1$.
Is this true?
If yes, is it possible to prove it from first principles, or is it something that needs some highly technical theorems?
If no, do we have an example of connected 1-dimensional manifold not homeomorphic to $\mathbb{R}$ or $S^1$?
By request, see here for an outline from first principles: http://www.igt.uni-stuttgart.de/eiserm/lehre/2014/Topologie/Gale%20-%201-manifolds.pdf
EDIT: this link is now dead. The reference is
This proof proves that all topological 1-manifolds (without boundary) are homeomorphic to $\mathbb{R}$ or $S^1$. From this we get the corollary that all $1$-manifolds can be endowed with $C^{\infty}$ structures. In fact, it is true that all 1-manifolds are diffeomorphic to $\mathbb{R}$ or $S^1$. For this proof, my guess is that using integral curves/orientations is probably the easiest, though check Guilliman/Pollack for sure.
Additionally, there is another proof with more of an algebraic topology flavor using CW complexes found in Lee's Topological Manifolds, Theorem 5.27.