Two very large numbers

The expofactorial function is like the factorial function but with exponents.

$\large n^{!}=n^{(n-1)^{!}}$

For example $4^{!}=4^{3^{2^{1}}}=4^{9}=262144$

Using Knuth's up arrow notation

$b \uparrow^{1} d = b^{d}$ , $b \uparrow^{n} 0 =1$ and $b \uparrow^{n} d = b \uparrow^{n-1} [b \uparrow^{n} (d-1)]$

For example $2 \uparrow^{2} 4 = 2 \uparrow^{1} [2 \uparrow^{2} 3] = 2 \uparrow [2 \uparrow [2 \uparrow^{2} 2]]=2 \uparrow [2 \uparrow [2 \uparrow [2 \uparrow^{2} 1]]] = 2^{2^{2^{2}}} = 65536$

Give the minimum integer of $n$ such that

$\huge 3 \uparrow^{n} 3 > 200^{!}$