Evaluating tetration to infinite heights (e.g., $2^{2^{2^{2^{.^{.^.}}}}}$)

Hint: Equations $y=x$ and $y=2^x$ do not intersect, means there's no solution for $x\in\mathbb R$.

Here are the first 20 solutions (for $\log(z)+k \cdot (2\cdot \pi\cdot i)$ such that $2^z=z$ or $z\cdot \log(2) = \log(z) + k\cdot(2 \pi i)$ : $$ \small \begin{array} {r|l} k & z : 2^z=z\\ \hline 0 & 0.82467854614+1.56743212385 \, î \\ 1 & 3.51523672192+10.8800532084 \, î \\ 2 & 4.36143141283+20.0871628060 \, î \\ 3 & 4.88885664543+29.2211855083 \, î \\ 4 & 5.27384865880+38.3277872288 \, î \\ 5 & 5.57736047492+47.4208762811 \, î \\ 6 & 5.82797084936+56.5062285608 \, î \\ 7 & 6.04142622483+65.5867042057 \, î \\ 8 & 6.22733941446+74.6638922661 \, î \\ 9 & 6.39201436790+83.7387507973 \, î \\ 10 & 6.53981045480+92.8118939379 \, î \\ 11 & 6.67386852707+101.883734634 \, î \\ 12 & 6.79652672953+110.954561406 \, î \\ 13 & 6.90957275012+120.024582285 \, î \\ 14 & 7.01440413644+129.093951274 \, î \\ 15 & 7.11213417750+138.162784952 \, î \\ 16 & 7.20366413122+147.231173287 \, î \\ 17 & 7.28973387243+156.299186877 \, î \\ 18 & 7.37095826487+165.366881942 \, î \\ 19 & 7.44785382986+174.434303834 \, î \end{array} $$

(Using Pari/GP , more than 100 digits precision)

l2=log(2)
pi2i = 2*Pi*I 
{list=matrix(20,2);
for(k=0,20-1,              \\ k contains branchno for logarithm
   x0=1+I;
   for(j=1,20,              \\ Newton-iteration
        x1=x0-(l2*x0-(log(x0)+k*pi2i))/(l2-(1/x0));
        if(abs(x1-x0)<1e-100,break(),x0=x1); );
   list[1+k,]=[k,x0];
 );}
 printp(list)

Any finite height tower can (in theory) be evaluated. It will equate to some natural number. If the tower is only moderately tall, it will be an enormous number.Wolfram Alpha shows that a tower only five layers high has $19729$ digits. If the tower height is infinite, the value diverges (quickly) to infinity and the value cannot be evaluated.

Your trick of equating to $x$ and substituting will find the limit if it exists. In this case, it does not.