How large is $f_{\Gamma_{0}}(2)$ in the Fast-Growing Hierarchy?

Question

If my ordinal arithmetic is correct, $f_{\Gamma_{0}}(2)=f_{\epsilon_{\epsilon_0}}(2)$ so $$f_{\epsilon_{\omega}}(2)\ll f_{\Gamma_{0}}(2)\ll f_{\epsilon_{\epsilon_1}}(2),$$ but that doesn't show anything new. Maybe bounds with Bower's Exploding Array Function or Cascading-E Notation can help.