Topological spaces with too many open sets

Consistently, $\omega^*$ has this property. In the paper

E. van Douwen, K. Kunen, and J. van Mill, "There can be $C^*$-embedded dense proper subspaces of $\beta\omega - \omega$," Proc. Amer. Math. Soc. 105 (1989), pp. 462-470, available here.

it was shown to be consistent that, for every $p \in \omega^*$, every bounded continuous real-valued function on $\omega^* - \{p\}$ can be extended to $p$. This almost gives a consistent answer to the OP's question, the only problem being that the property in the question does not mention boundedness. However, in the context of $\omega^*$ it turns out that adding in boundedness does not hurt anything:

Observation: If $p \in \omega^*$, then every real-valued continuous function on $\omega^* - \{p\}$ is bounded.

Proof: Suppose $f$ is an unbounded continuous real-valued function on $\omega^* - \{p\}$. Pick a sequence $\langle x_n : n < \omega \rangle$ of points in $\omega^* - \{p\}$ such that $f(x_n)$ converges to infinity. It is well known that $\omega^*$ is compact and contains no nontrivial convergent sequences. Thus there must be a point $q \in \omega^*$ with $q \neq p$ at which the $x_n$ cluster. Since $f(q)$ is some (finite) real number but $f(x_n)$ goes to infinity, this contradicts the continuity of $f$. $\qquad$ QED

On the other hand, Fine and Gillman proved that $\omega^*$ consistently fails to have the OP's property (CH implies that it does not).

I do not know whether your question has a positive answer in ZFC (but I suspect that it does).

Here is a link to a related paper that you might find interesting (it's where I learned about the two results I mention above).

Any extremally disconnected space without isolated points has this property because a space is extremally disconnected if and only if every dense subset is C*-embedded. (This result is in problem 6M of Gillman and Jerison.)