playing with power tower and nested radicals I get :
Prove that
Let $a_1=\sqrt{2}$ ,$a_2=\sqrt{2}^{\sqrt{2}}$,$a_3=\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}$,$a_4=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}}$ and $a_5=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}}}$ then we have : $$\sqrt{1+a_1\sqrt{1+a_2\sqrt{1+a_3\sqrt{1+a_4\sqrt{1+a_5}}}}}<2$$
You can check the value here. This is an almost integer .
I have tried to find a recursion formula without success .If we continue like this we get a limit by the Herschfeld's theorem . Moreover I have tried brut force but it's not nice the power series by example the power series of $f(x)=\sqrt{x}^{\sqrt{x}}$ at $x=2$.And finally I have tried to apply something near to the Somos answer without success again .
If you have nice idea to prove it .
Thanks a lot for all your contributions !