-2
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Fifth Submission
Python 3.8 (pre-release), 175 bytes
f=lambda x,z,y:y+z if x==0 else eval(f"f({x-1},{z},"*(a:=eval(f"f({x-1},{z},"*z+f"{y}"+")"*z))+f"{y}"+f")"*a)
n=9**9
for _ in range(f(*[99**99]*3)):
n=f(n,n,n)**_**_
print(n)
Try it online!
Explanation coming later.