What is a 100th percentile? I am really confused behind the mathematical meaning of a 100th percentile.
What does that mean mathematically? Does it mean that a data point in a sample space is greater in some metric and that is also greater than itself?
That makes no sense. AFAIK, there can be no such thing as the 100th percentile, because the maximum data point considered is still part of the sample space.
For example, when a student scores the highest marks, he/she can be the 99.999th percentile, but what is the meaning of 100th percentile ?
 A: I cannot comment there. So I am posting an answer.
When the sample space is infinitely large as compared to the point/interval you are referring to, then 100 percentile makes sense.
For instance, the value of $y(0)$ in the function $e^{-x^2}$ is a 100 percentile.
A: Before attempting to give an answer to the OP, I briefly discuss the definition of a quantile (percentile is also used)

Definition:

*

*Given a (cumulative) distribution $F$ on $\mathbb{R}$, for $0<q<1$
a $q$-quantile (or percentile) is a number $z_q$ such that
$$
\begin{align}
F(z_q-)\leq q\leq F(z_q)\tag{0}\label{quantile}
\end{align}
$$

*Similarly, if $X$ is a real valued random variable with distribution $F_X$, for $0<q<1$ a $q$-quantile is a number $z_q$ such that
$$
\begin{align}\mathbb{P}[X<z_q]=F_X(z_q-)\leq q\leq F(z_q)=\mathbb{P}[X\leq q]\tag{1}\label{quantileX}
\end{align}
$$

Some special quantile functions are

*

*Consider the functions $Q$, $Q^+$ defined on $(0,1)$ by
$$ \begin{align}
Q(q)&=\inf\{x\in\mathbb{R}: F(x)\geq q\}\\
Q^+(q)&=\sup\{x\in\mathbb{R}:F(x-)\leq q\}
\end{align}
$$
It is not difficult to check that $Q(q)$ and $Q^+(q)$ satisfy \eqref{quantile} for each $0<q<1$. $Q$ and $Q_+$ have the additional  property that for any other $q$-quantile  $z_q$ of $F$ satisfies
$$Q(q)\leq z_q\leq Q^+(q)$$

Back to the OP:
From the definition of a quantile, it is easy to see that

*

*Unless $F$ (or rather the Probability measure induced by $F$) is supported an a interval bounded from below, the definition of $q$-quantile cannot be expended to $q=0$.

*Unless $F$ is supported on an interval bounded from above, the definition of $q$-quantile cannot be extended to $q=1$.

However


*If $F$ is supported in an interval bounded from below, one can extend the notion of  $q$-quatile with $q=0$ can be done, and the functions $Q$ and $Q^+$ can de defined at $q=0$ as extended real numbers $Q(0)=-\infty<Q^+(0)=\sup\{x:F(x-)=0\}$


*If $F$ is supported in an interval bounded from above, one can extend the notion of  $q$-quatile with $q=1$, and   the functions $Q$ and $Q^+$ can de defined at $q=1$ as extended real numbers $Q(1)=\in\{x:F(x-)=1\}<\infty=Q^+(1)$
For sample of size $n$ of some distribution $F$ on the real line, say $x_1,\ldots, x_n$ one typically considers the empirical distribution $$F_n(x)=\frac{1}{n}\sum^n_{k=1}\mathbb{1}_{(-\infty,x]}(x_k)$$
Of course $F_n$ is supported in the compact interval interval $[\min_kx_k,\max_kx_k]$, so there $0$ and $1$ quantiles are mathematically well deinided in this situation. There is no harm to define the $0$-quantile and the $1$-quantile of $F_n$ (or the sample X) as $\min_kx_k$ and $\max_kx_m$ respectively.  In fact, many statistical packages display the $0$ and $1$ quantiles as the min and the max of the sample.
