User blog comment:Primussupremus/Another attempt at defining some type of notation./@comment-28606698-20170419183525

I would say

$$5e5\approx f_{\omega}(5)$$

$$5e5\# 5\approx f_{\omega+1}(5)$$

$$5e5\# (5)\approx f_{\omega+2}(5)$$

$$5e5\# (5e5\# (5))\approx f_{\omega+2}^2(5)$$

bigger numbers seem are not well-defined.