Prime Exponentials

I assume the number above $19$ is $17$ as you have listed the first $10$ prime numbers in ascending order.

The trick is to apply Euler's Totient Function repeatedly

$\varphi(100) = 40, \varphi(40) = 16, \varphi(100) = 8$

And $\text{gcd} (29,100) = \text{gcd} (23,40) = \text{gcd} (19,16) = \text{gcd} (17,8) = 1$

$\begin{aligned} \LARGE 29^{23^{19^{17^{.^{.^{7^{5^{3^2}}} }}}}} & \equiv & 29^{23^{19^{17^{.^{.^{7^{5^{3^2}}} }}}}} \pmod {100} \\ & \equiv & 29^{23^{19^{17^{.^{.^{7^{5^{3^2}}} }}}} \pmod {40} } \pmod {100} \\ & \equiv & 29^{23^{19^{17^{.^{.^{7^{5^{3^2}}} }}} \pmod {16} } \pmod {40} } \pmod {100} \\ & \equiv & 29^{23^{19^{17^{.^{.^{7^{5^{3^2}}} }} \pmod {8} } \pmod {16} } \pmod {40} } \pmod {100} \\ & \equiv & 29^{23^{19^{1 } \pmod {16} } \pmod {40} } \pmod {100} \\ & \equiv & 29^{23^3 \pmod {40} } \pmod {100} \\ & \equiv & 29^7 \pmod {100} \\ & \equiv & \boxed{9} \\ \end{aligned}$