User blog comment:Alemagno12/Fast-Growing Functions in Lambda Calculus (and their implementation in CoC)/@comment-26454151-20171217192234/@comment-26454151-20171220003527

I was going to say that the Goodstein sequence can be programmed in CoC, but I had a revelation while typing that response.

In untyped lambda calculus, it's a fairly trivial matter to do as it is Turing complete and is able to perform general recursion. Inside Barendregt's lambda cube, however, there is a restriction on recursion and there is no access to fixed-point combinators. In this typed lambda calculus, there is a way to define ordinals. This ordinal definition allows you to construct limit ordinals from a lambda function  which describes the fundamental sequence of an ordinal.

I found a way to express the Hardy hierarchy using this definition of ordinals, and since exponentiation can be easily defined in this system of ordinals, we can actually reach $$\varepsilon _0$$ quite easily. For that matter, we can even reach $$\zeta _0$$ pretty easily as well. This is all pretty recent, so I haven't yet worked out exactly how far this method can be taken.