Looking at the wiki article for the Stone-Cech compactification, it states:
Andrey Nikolayevich Tikhonov introduced completely regular spaces in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps.
Completely regular spaces are T1 spaces where points can be separated from closed sets by continuous maps into the interval $[0, 1]$, so these of course have non-trivial continuous functions.
My question is - what is an example of such a pathological space as above? E.g. an example of a Hausdorff space whose only continuous real-valued functions are constant maps? The source cited ultimately ends up being Tikhonov's 1930 paper in Russian, which I unfortunately cannot read.
Spaces for which every continuous function $f:X\to\mathbb{R}$ is constant are called strongly connected.
Pi-base is an useful database of topological spaces that lists some of them, see here. Also see the classic book, "Counterexamples in topology" by Steen and Seebach.
One of the more geometric examples here seems to be the irrational slope topology, see here. The reason given in the database is that its connected and countable, hence automatically strongly connected.