(Intuition) how big is the church-kleene ordinal?

Question

The church kleene ordinal is defined as the first ordinal that cannot be computed.

But how should we imagine this ordinal? What is the "limit process" that leads to it?

Answer 1

$\omega_1^{CK}$ is also the first uncomputable ordinal. So if $f(n)$ is the largest ordinal defined by a length $n$ turingmachine. Then $\omega_1^{CK}$ is $$\lim_{n\to\omega} f(n)$$