Up arrow notation: concept check

Using Knuth's up arrow notation for large numbers:

$\displaystyle m \uparrow\uparrow 6 = m^{m^{m^{m^{m^{m}}}}}$

otherwise referred to as a tower of m's with 6 floors. My question is in regards to the analogy; if I continuously iterate the $\displaystyle \uparrow$ for $\displaystyle m \uparrow\uparrow n$operator, am I taking that tower and placing a copy of it on every floor of another tower of n floors?

As in $\displaystyle 3 \uparrow\uparrow\uparrow 2 = 3^{27} \Rightarrow 3 \uparrow\uparrow\uparrow\uparrow 2 = (3^{27})^{(3^{27})}$ ?

Also would the following simplification work out? $\displaystyle 5 \uparrow\uparrow\uparrow\uparrow 3 = 5 \uparrow\uparrow\uparrow 3^{5 \uparrow\uparrow\uparrow 3^{5 \uparrow\uparrow\uparrow 3}}?$