For a book-length answer to your question, see Statistical Modelling with Quantile Functions by Warren Gilchrist. In the following we assume random variables are continuous, let $F$ the the cdf of $X$, then the quantile function is the inverse of $F$, $$ Q(p) = F^{-1}(p), p \in [0, 1] $$ $Q$ is then a monotonously increasing function, and importantly, any monotonously increasinf function of argument $p\in [0,1]$ is the quantile function of some (real) random variable. That makes it easy to construct new quantile function from existing ones, making for flexibility in modeling, and we can use that to easily make some quantile functions that do not have analytical inverses.

Some examples:
Sums of quantile functions are quantile functions
Sums of monotonously increasing functions are also monotonously increasing

Quantile functions to a positive power
A positive power $Q^\alpha$ of an increasing function is increasing

Convex combinations of quantile functions
If $Q_1, Q_2$ are increasing, then so is $\omega Q_1 + (1-\omega) Q_2$ for $0\le \omega \le 1$

Positive linear combinations of quantile functions
If $Q_1, Q_2$ are increasing, so is $a Q_1 + b Q_2$ for $a\ge 0, b\ge 0$

Products of quantile functions corresponding to non-negative random variables
If $Q_1 \ge 0, Q_2 \ge 0$ then so is $Q_1 Q_2 \ge 0$. This also works if only one of the quantile functions correspond to a non-negative random variable.

(Some of this conditions can be summarized by saying that the set of quantile functions constitute a convex cone)

So for some examples we can construct using this.

The (standard) exponential distribution have quantile function $Q(p)=-\log(1-p)$. If we reflect this distribution in the $y$-axis, the quantile function becomes $\log p$. The sum of this quantile functions is $$ \log(p) - \log(1-p) = \log\left( \frac{p}{1-p} \right)$$ which is the quantile function of the logistic distribution. If we take a convex combination, $$\alpha \log p - (1-\alpha) \log(1-p)$$ we get a skew distribution, the skew logistic, which do not have an analytic cdf.

The quantile function of the standard uniform random variable is $Q(p)=p$. We can combine this linearly with the logistic to get $$Q(p)= \log\left( \frac{p}{1-p} \right) + Kp, K>0$$ which will have the effect of flattening the maximum. This quantile function can not be inverted analytically, either. Now you have the tools to construct many examples yourself!

(I will add some plots later)

For a book-length answer to your question, see Statistical Modelling with Quantile Functions by Warren Gilchrist. In the following we assume random variables are continuous, let $F$ the the cdf of $X$, then the quantile function is the inverse of $F$, $$ Q(p) = F^{-1}(p), p \in [0, 1] $$ $Q$ is then a monotonically increasing function, and importantly, any monotonically increasing function of argument $p\in [0,1]$ is the quantile function of some (real) random variable. That makes it easy to construct new quantile function from existing ones, making for flexibility in modeling, and we can use that to easily make some quantile functions that do not have analytical inverses.

Some examples:
Sums of quantile functions are quantile functions
Sums of monotonically increasing functions are also monotonically increasing

Quantile functions to a positive power
A positive power $Q^\alpha$ of an increasing function is increasing

Convex combinations of quantile functions
If $Q_1, Q_2$ are increasing, then so is $\omega Q_1 + (1-\omega) Q_2$ for $0\le \omega \le 1$

Positive linear combinations of quantile functions
If $Q_1, Q_2$ are increasing, so is $a Q_1 + b Q_2$ for $a\ge 0, b\ge 0$

Products of quantile functions corresponding to non-negative random variables
If $Q_1 \ge 0, Q_2 \ge 0$ then so is $Q_1 Q_2 \ge 0$. This also works if only one of the quantile functions correspond to a non-negative random variable.

(Some of this conditions can be summarized by saying that the set of quantile functions constitute a convex cone)

So for some examples we can construct using this.

The (standard) exponential distribution have quantile function $Q(p)=-\log(1-p)$. If we reflect this distribution in the $y$-axis, the quantile function becomes $\log p$. The sum of this quantile functions is $$ \log(p) - \log(1-p) = \log\left( \frac{p}{1-p} \right)$$ which is the quantile function of the logistic distribution. If we take a convex combination, $$\alpha \log p - (1-\alpha) \log(1-p)$$ we get a skew distribution, the skew logistic, which do not have an analytic cdf.

The quantile function of the standard uniform random variable is $Q(p)=p$. We can combine this linearly with the logistic to get $$Q(p)= \log\left( \frac{p}{1-p} \right) + Kp, K>0$$ which will have the effect of flattening the maximum. This quantile function can not be inverted analytically, either. Now you have the tools to construct many examples yourself!

(I will add some plots later)

For a book-length answer to your question, see Statistical Modelling with Quantile Functions by Warren Gilchrist. In the following we assume random variables are continuous, let $F$ the the cdf of $X$, then the quantile function is the inverse of $F$, $$ Q(p) = F^{-1}(p), p \in [0, 1] $$ $Q$ is then a monotonously increasing function, and importantly, any monotonously increasinf function of argument $p\in [0,1]$ is the quantile function of some (real) random variable. That makes it easy to construct new quantile function from existing ones, making for flexibility in modeling, and we can use that to easily make some quantile functions that do not have analytical inverses.

Some examples:
Sums of quantile functions are quantile functions
Sums of monotonously increasing functions are also monotonously increasing

Quantile functions to a positive power
A positive power $Q^\alpha$ of an increasing function is increasing

Convex combinations of quantile functions
If $Q_1, Q_2$ are increasing, then so is $\omega Q_1 + (1-\omega) Q_2$ for $0\le \omega \le 1$

Positive linear combinations of quantile functions
If $Q_1, Q_2$ are increasing, so is $a Q_1 + b Q_2$ for $a\ge 0, b\ge 0$

Products of quantile functions corresponding to non-negative random variables
If $Q_1 \ge 0, Q_2 \ge 0$ then so is $Q_1 Q_2 \ge 0$. This also works if only one of the quantile functions correspond to a non-negative random variable.

So for some examples we can construct using this.

The (standard) exponential distribution have quantile function $Q(p)=-\log(1-p)$. If we reflect this distribution in the $y$-axis, the quantile function becomes $\log p$. The sum of this quantile functions is $$ \log(p) - \log(1-p) = \log\left( \frac{p}{1-p} \right)$$ which is the quantile function of the logistic distribution. If we take a convex combination, $$\alpha \log p - (1-\alpha) \log(1-p)$$ we get a skew distribution, the skew logistic, which do not have an analytic cdf.

The quantile function of the standard uniform random variable is $Q(p)=p$. We can combine this linearly with the logistic to get $$Q(p)= \log\left( \frac{p}{1-p} \right) + Kp, K>0$$ which will have the effect of flattening the maximum. This quantile function can not be inverted analytically, either. Now you have the tools to construct many examples yourself!

(I will add some plots later)

For a book-length answer to your question, see Statistical Modelling with Quantile Functions by Warren Gilchrist. In the following we assume random variables are continuous, let $F$ the the cdf of $X$, then the quantile function is the inverse of $F$, $$ Q(p) = F^{-1}(p), p \in [0, 1] $$ $Q$ is then a monotonously increasing function, and importantly, any monotonously increasinf function of argument $p\in [0,1]$ is the quantile function of some (real) random variable. That makes it easy to construct new quantile function from existing ones, making for flexibility in modeling, and we can use that to easily make some quantile functions that do not have analytical inverses.

Some examples:
Sums of quantile functions are quantile functions
Sums of monotonously increasing functions are also monotonously increasing

Quantile functions to a positive power
A positive power $Q^\alpha$ of an increasing function is increasing

Convex combinations of quantile functions
If $Q_1, Q_2$ are increasing, then so is $\omega Q_1 + (1-\omega) Q_2$ for $0\le \omega \le 1$

Positive linear combinations of quantile functions
If $Q_1, Q_2$ are increasing, so is $a Q_1 + b Q_2$ for $a\ge 0, b\ge 0$

Products of quantile functions corresponding to non-negative random variables
If $Q_1 \ge 0, Q_2 \ge 0$ then so is $Q_1 Q_2 \ge 0$. This also works if only one of the quantile functions correspond to a non-negative random variable.

(Some of this conditions can be summarized by saying that the set of quantile functions constitute a convex cone)

So for some examples we can construct using this.

The (standard) exponential distribution have quantile function $Q(p)=-\log(1-p)$. If we reflect this distribution in the $y$-axis, the quantile function becomes $\log p$. The sum of this quantile functions is $$ \log(p) - \log(1-p) = \log\left( \frac{p}{1-p} \right)$$ which is the quantile function of the logistic distribution. If we take a convex combination, $$\alpha \log p - (1-\alpha) \log(1-p)$$ we get a skew distribution, the skew logistic, which do not have an analytic cdf.

The quantile function of the standard uniform random variable is $Q(p)=p$. We can combine this linearly with the logistic to get $$Q(p)= \log\left( \frac{p}{1-p} \right) + Kp, K>0$$ which will have the effect of flattening the maximum. This quantile function can not be inverted analytically, either. Now you have the tools to construct many examples yourself!

(I will add some plots later)