What does a number look like that's in $\mathbb{Z}\left[\sqrt{14}, \frac{1}{2}\right]$ that's not in $\mathbb{Z}[\sqrt{14}]$?

An element of this ring can be considered a polynomial in $\frac12$, with coefficients in $\mathbf Z[\sqrt{14}]$. Reducing all terms to the same denominator, an element can ultimately be written as $$\frac{a+b\sqrt{14}}{2^n} \quad (a,b\in\mathbf Z).$$

For readers aware of localisations, it also may be described as the ring of fractions of $\mathbf Z[\sqrt{14}]$ w.r.t. the multiplicative subset of powers of $\frac12$.

If $a,b \in \mathbb R$, then $\mathbb{Z}[a,b]$ is the smallest ring that contains $a,b$; it is the same as the set of all polynomial expressions in $a,b$ with coefficients in $\mathbb{Z}$.

When $a=\sqrt{14}$ and $b=1/2$, all powers of $a$ reduce to an integer or to an integer times $a$. There is no reduction for powers of $b$, but all fractions can be reduced to the same denominator.

Therefore, a typical element of $\mathbb{Z}\left[\sqrt{14}, \frac{1}{2}\right]$ is of the form $\dfrac{u+v\sqrt{14}}{2^n}$, for $u,v \in \mathbb Z$ and $n \in \mathbb N$.