Let's say that you have a distribution $F$ in the exponential family with density
\begin{align}
\newcommand{\mbx}{\mathbf x}
\newcommand{\btheta}{\boldsymbol{\theta}}
f(\mbx \mid \btheta) &= \exp\bigl(\eta(\btheta) \cdot T(\mbx) - g(\btheta) + h(\mbx)\bigr)
\end{align}

Given independent realizations $\{x_1, x_2, \dotsc, x_n\}$ of $F$ (with unknown parameter $\theta$), then the distribution over $\theta$, $F'$, is the conjugate prior of $F$. The density of $F'$ is
\begin{align}
f(\btheta \mid \boldsymbol\phi) = L(\btheta \mid \mbx_1, \dotsc, \mbx_n)
&= f(\mbx_1, \dotsc, \mbx_n \mid \btheta) \\\\
&\propto \prod_i f(\mbx_i\mid \btheta) \\\\
&= \textstyle\prod_i\exp\Bigl(\eta(\btheta) \cdot \textstyle T\left(\mbx_i\right) - g(\btheta) + h(\mbx_i)\Bigr) \\\\
&\propto \textstyle\prod_i\exp\Bigl(\eta(\btheta) \cdot \textstyle T\left(\mbx_i\right) - g(\btheta)\Bigr) \\\\
&= \textstyle\exp\Bigl(\eta(\btheta) \cdot \bigl(\textstyle\sum_iT\left(\mbx_i\right)\bigr) - ng(\btheta)\Bigr) \\\\
&= \exp\bigl(\eta'(\boldsymbol \phi) \cdot T'(\btheta)\bigr)
\end{align}
where
\begin{align}
\eta'(\boldsymbol\phi) &=
\begin{bmatrix}
\sum_iT_1(\mbx_i) \\\\
\vdots \\\\
\sum_iT_k(\mbx_i) \\\\
\sum_i1
\end{bmatrix}
&
T'(\btheta) &=
\begin{bmatrix}
\eta_1(\btheta) \\\\
\vdots \\\\
\eta_k(\btheta) \\\\
-g(\btheta)
\end{bmatrix}.
\end{align}
Thus, $F'$ is also in the exponential family ($T'$ replaced $\eta$ and $\eta'$ replaced $T$ since this distribution is over $\theta$ the parameter of the distribution over $x$.)

Interestingly, $\boldsymbol\phi$ has exactly one more parameter than $\btheta$ except in the rare case where natural parameter $\phi_{k+1}$ is redundant, but such a distribution would be very weird (it would mean that the number of observations $\mbx$, that is, $n$, tells you nothing about $\btheta$.)

So, to answer your question, with each conjugate prior you get exactly one more hyperparameter.

There are many conjugate priors of the Gaussian distribution depending on how you look at it. In my opinion, the analogy to the Multinomial-Dirichlet example would set things up as follows: assume that $n$ real-valued numbers are generated by a Gaussian with unknown mean and variance. Then, the distribution of the mean and variance given the data points is a three-parameter conjugate prior distribution whose sufficient statistics are the total of the samples, the total of the squares of the samples, and the number of samples.