3 bridge

The given expression is :

$\large 3^{3^3 \times 3} \times 3^{3^3 \times 3} \times 3^{3^3 \times 3} \times 3 = 3^a$

Now, $3^3 = 27$

$\implies \large 3^{27 \times 3} \times 3^{27 \times 3} \times 3^{27 \times 3} \times 3^1 = 3^a$

$\implies \large 3^{81} \times 3^{81} \times 3^{81} \times 3^1 = 3^a$

$\implies \large 3^{81 + 81 + 81 + 1} = 3^a$

$\implies \large 3^{244} = 3^a$

$\implies \large a = 244$

$\color{#20A900}\text{Therefore, sum of digits of a = 2 + 4 + 4 = 10}$

$\begin{array}{rl} \large 3^{(3^{3} \times 3)} \times 3^{(3^{3} \times 3)} \times 3^{(3^{3} \times 3)} \times 3 &= \ \ \large 3^{(3^{4})} \times 3^{(3^{4})} \times 3^{(3^{4})} \times 3 \\ &= \ \ \large 3^{(3^{4} + 3^{4} + 3^{4})} \times 3 \\ &= \ \ \large 3^{(3^{4} \times 3)} \times 3 \\ &= \ \ \large 3^{(3^{5})} \times 3 \\ &= \ \ \large 3^{(3^{5} + 1)} \ = \ 3^a \\ \end{array}$

Thus $a = 3^5 + 1 = 244$ , so our answer is $2 + 4 + 4 = \boxed{10}$