Edit: I've added some information about the game with a restricted number $N$ of rounds, since the OP mentioned this.
Just a comment to complement the nice answers we have already.
There is an algorithm that calculates the optimal strategy and the value of each state for such an optimal stopping problem.
Let $f(x)$ be the payout at each state $x\in {\cal S}$ for the underlying Markov chain $(X_n)$, and let $g(x)$ be the cost of leaving state $x$.
Set $u_0=f$ and for $N\geq 1$ put $u_N=\max(Pu_{N-1}-g,f)$. Here $P$ is the transition matrix of the Markov chain. Then $u_N$ is the value function for the restricted game with at most $N$ rounds; that is, $u_N(x)=\sup_{T\leq N}\ \mathbb{E}\left[ f(X_T)\, | \, X_0=x\right],$ where the supremum is over all stopping times $T$ that satisfy $T\leq N$.
As $N\to\infty$, we get $u_N\uparrow v$ where $v$ is the value function for the unrestricted game. The optimal strategy is to stop when the chain hits the set $\lbrace x: f(x)=v(x)\rbrace$. Here $v(x)$ means the value of state $x$, i.e., $v(x)$ is the maximum expected payout starting in state $x$ and using any finite stopping time $T$ as a strategy:
$v(x)=\sup_T\ \mathbb{E}\left[ f(X_T)\, | \, X_0=x\right].$
This is all nicely explained in the section on Optimal Stopping in Gregory Lawler's book "Introduction to Stochastic Processes".
In your problem we have $f(x)=x$ for $1\leq x\leq 100$, and $g(x)\equiv 1$. Your Markov chain $(X_n)$ is a sequence of independent, uniform $\lbrace 1,2,\dots, 100\rbrace $ random variables. Thus the $P$ operator applied to any function $h$ gives the constant function whose value is the average value of $h$, that is, $Ph(x)\equiv \sum_{i=1}^{100} h(i)/100$. So we calculate $u_0(x)=x,\quad u_1(x)=\max\left(x,{99\over 2}\right), \quad u_2(x)=\max\left(x,{12301\over 200}\right). $
Taking very large $N$ gives $v(x)\approx \max\left(x,{86.35714}\right),$ which shows that the optimal strategy (over all finite stopping times!) is to quit as soon as we get $87$ or higher, and the value of the game is $86.35714$ dollars.
In your problem it is pretty straightforward to calculate the exact answer, but this algorithm also gives the answer for more complicated games where exact calculations are not so easy.