Last two digits

We note that $7^7 \equiv 7^47^3 \equiv 49^2 7^3 \equiv (50-1)^27^3 \equiv 7^3 \equiv 343 \equiv 43 \text{ (mod 100)}$ ; $\implies 7^7 \equiv 100n + 43$ , where $n$ is an integer.

Now consider $7^{7^7} \equiv 7^{100n+43} \equiv 49^{50n+20}7^3 \equiv (50-1)^{50n+20}7^3 \equiv 7^3 \equiv 343 \equiv 43 \text{ (mod 100)}$ .

Similarly, $\large 7^{7^{7^{\cdot ^{\cdot ^\cdot}}}} \equiv \boxed{43} \text{ (mod 100)}$ .

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Alternative solution

$\large \begin{aligned} 7^{7^{7^{7^{7^{7^7}}}}} & \equiv 7^{7^{7^{7^{7^{7^7}}}} \color{#3D99F6} \text{ mod }\phi(100)} \text{ (mod 100)} & \small \color{#3D99F6} \text{As }\gcd(7,100)=1 \text{, Euler's theorem applies.} \\ & \equiv 7^{7^{7^{7^{7^{7^7}}}} \color{#3D99F6} \text{ mod }40} \text{ (mod 100)} & \small \color{#3D99F6} \text{Euler's totient function }\phi(100) = 40 \\ & \equiv 7^{7^{7^{7^{7^{7^7\text{ mod }2}\text{ mod }4}\text{ mod }8}\text{ mod }16} \text{ mod }40} \text{ (mod 100)} & \small \color{#3D99F6} \text{Repeated use of Euler's theorem.} \\ & \equiv 7^{7^{7^{7^{\color{#3D99F6}7^1 \text{ mod }4} \text{ mod }8} \text{ mod }16} \text{ mod }40} \text{ (mod 100)} & \small \color{#3D99F6} 7 \equiv 3 \text{ (mod 4)} \\ & \equiv 7^{7^{7^{\color{#3D99F6}7^3 \text{ mod }8} \text{ mod }16} \text{ mod }40} \text{ (mod 100)} & \small \color{#3D99F6} 7^3 \equiv 7\cdot 49 \equiv 7(48+1) \equiv 7 \text{ (mod 8)} \\ & \equiv 7^{7^{\color{#3D99F6}7^7 \text{ mod }16} \text{ mod }40} \text{ (mod 100)} & \small \color{#3D99F6} 7^7 \equiv 7\cdot 49^3 \equiv 7(48+1)^3 \equiv 7 \text{ (mod 16)} \\ & \equiv 7^{\color{#3D99F6}7^7 \text{ mod }40} \text{ (mod 100)} & \small \color{#3D99F6} \text{See note: } 7^7 \equiv 23 \text{ (mod 16)} \\ & \equiv 7^{23} \text{ (mod 100)} \\ & \equiv \boxed{43} \text{ (mod 100)} & \small \color{#3D99F6} \text{See note} \end{aligned}$

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Notes:

$7^7 \equiv 7\cdot 49^3 \equiv 7(40+9)^3 \equiv 7\cdot 9^3 \equiv 7\cdot 81 \cdot 9 \equiv 7\cdot 9 \equiv 63 \equiv 23 \text{ (mod 40)}$

$7^{23} \equiv 7\cdot (49)^{11} \equiv 7 \cdot 49(50-1)^{10} \equiv 7\cdot 49 \equiv 343 \equiv 43 \text{ (mod 100)}$