Find the value of $0.5^{0.5^{0.5^{0.5^\ldots}}}$
My attempt: $$\begin{aligned} &0.5^{0.5^{0.5^{0.5^\ldots}}} =: x \\ &\Rightarrow\ 0.5^x = x \\ &\Rightarrow\ -x \ln(2) = \ln(x) \\ &\Rightarrow\ \ln(x) = -x \ln(2) \\ &\Rightarrow x\approx0.64 \end{aligned}$$
But, by letting the value is $x$ means I am assuming that the expresstion has a real value $x$. How we can prove that more logically.
To summarize the discussion in the comments:
The confusion is a result of the fact that iterated exponentiation is non-associative. When one writes, e.g., $a^{b^c}$, what is meant is $a^{(b^c)}$, not $\left(a^b\right)^c$. The latter is just $a^{bc}$, of course.
The OP was, apparently, computing the iterated square root $$\cdots \sqrt {\sqrt {\sqrt {.5}}}$$ which is indeed $1$.
However, this is not what is meant by the original iterated exponentiation. That problem can be rewritten as $\lim_{n\to \infty} a_n$ where the $a_n$ are defined recursively as $$a_1=.5\quad a_n=.5^{a_{n-1}}$$
If you use that definition you get the desired value very rapidly.