I search a composite number near $4^{4^4}$ with a very large smallest prime factor. A candidate is $$4^{4^4}+253=4^{256}+253$$
The number is composite and has $155$ digits, so it is in the range , the quadratic sieve can handle. But the calculation will take several days.
I ran $500$ elliptic curves with $B1=250K$ and $150$ curves with $B1=1M$ and will continue the search with ECM. Everyone reading this is invited to run MPQS.