Answer: 61

Small Hint:

For a prime $p>50,$ compare the exponent of $p$ in $200!$ with twice its exponent in $100!$

Big Hint:

Primes from $67$ through $99$ cancel; primes from $51$ through $66$ do not

Solution:

For a prime $p>50,$ we have $p^2>200,$ so $$\begin{aligned} v_p\left(\binom{200}{100}\right) &=\left\lfloor\frac{200}{p}\right\rfloor\\ &\quad-2\left\lfloor\frac{100}{p}\right\rfloor. \end{aligned}$$ If $67\leq p<100,$ this is $2-2=0.$ If $50<p\leq66,$ it is $3-2=1.$ Hence no prime greater than $66$ divides the binomial coefficient, while every prime between $50$ and $66$ does. The largest such prime is $61.$