In a bid to make this even more complex as from a bit of searching this all seems to have been investigated before, [almost always true!].
We can use the notation to construct even more intractable numbers such as \(1;(1;n)\), for \(n \in \mathbb{Z}\). However, these are in themselves, repdigits, i.e just \(1;1\), \(1;11\), \(1;111\).
Of course, numbers such as \(1;(1;19)\),\(1;(1;23)\) have a chance of being prime also!