Nature of primes as the building blocks of integers?

It is considered standard in mathematics that all integers can be expressed as the product of primes:
$$n=p_1^{a_1}p_2^{a_2}...p_k^{a_k}$$
Where $p_i$ is prime and $p_{i-1}

Answer

Yes, of course your assumption is correct.

Consider for example, $\large{n=3^{56}\cdot 5^{48}}.$

Let us colour exponents that are composite, and indent for each level of tetration for which the action must be repeated:

\begin{align}
&\quad n&=&\quad 3^{\color\red {56}}\cdot 5^{\color\red {48}}\\
\text{tetration level 1:}&\quad \color\red {56}&=&\quad 2^3\cdot 7\\
&\quad \color\red {48}&=&\quad 3\cdot 2^{\color\red {4}}\\
\text{tetration level 2:}&\quad \color\red {4}&=&\quad 2^2\\
\Rightarrow&\quad \large{n}&=&\quad \large{3^{2^{3}\cdot 7}\cdot 5^{3\cdot 2^{2^2}}}\\
\end{align}

If it were not true (ie, if there were some stage at some level of tetration that factorisation yielded neither prime nor composite), it would of course contradict the fundamental theorem of arithmetic.