I checked the prime factors of
$$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$
with trial division and found non below $8*10^9$
Nevertheless, the given number has still a very low chance to be prime. Can anything be said about the possible prime factors ? The only restriction I found is that -3 must be a quadratic residue for any prime factor.
You can join in searching for a prime factor at github :