Arithmetic-Exponential Progression!

We can see that $1.1^{1.3^{1.5 \cdots}} = \infty$

So, our value above becomes $\large 0.5^{0.7^{0.9^{\infty}}}$

Now, since $x^\infty = 0$ if $0 \leq x<1$ , then our value simplifies to $\large 0.5^{0.7^0}$

Since $x^0=1$ for $x \neq 0$ , then our value simplifies further to $\large 0.5^1$

Now, since $\large 0.5^1 = 0.5$ , then our answer is $\boxed{0.5}$