The largest uncontested number is currently Deedlit's. There are two potentially larger numbers, but they are floating around at sizes where you can't hand-wave things that are being hand-waved, and I'll let someone with a sufficient math degree suss that out.

Instead, let's add more chaos. I'm going to propose a very large number, but I don't know how large. I solved a problem that was bugging me about an idea I had for a suggestion.

Worms come in orders now.

An order 0 worm is just a regular worm.

An order k worm is a list of order k-1 worms.

The local head of an order k worm is the highest indexed element of the order k worm, the global head is the local head chosen recursively until you have an integer.

The position of each integer in a worm can be specified as a 1-indexed vector based on the indicies from the groups it is a member of.

For example, in the worm 0101|1

010|11 the bolded 0 has a position <2:2>

If the global head is 0, then to decrement an order k worm you:

1) Decrement the local head

- if you are in an order 0 worm, chop off the 0, skip the remaining steps, and return the new length l as the vector <l>

- if you are in an order k > 0 worm, decrement the local head to obtain an inspection vector

v2) Duplicate

- let l be the length of the current order k worm, form the inspection vector l:

v, decrease l until you reach the first (IE highest) k+1:

v such that k:

v is an empty cell. Duplicate indicies k+1 through l n times. Return the vector k+1:

vIf the global head is greater than 0, then to decrement an order k worm you:

1) Decrement the local head

- if you are in an order 0 worm, grow it like a normal worm, return the length d of the dead worm as the vector <d+1>, skip the remaining steps

- if you are in an order k worm, let l be the length of the current order k worm, decrement the local head to obtain inspection vector l:

v.

2) Grow

- decrease l until you reach the first (IE highest) k+1:

v such that element k:

v is less than the current (IE post decrement)

global head, or it's empty. Create l-k-1 copies of element l and append them to the current order k worm, which is now of length l2, let d be the first component of vector

v, then, for elements k+1 through l-1, slice from element d through to the end of that order k-1 worm and append them pairwise to elements l through to l2-1, return the vector k+1:

vThe goal was to embed a Buchholz hydra into a 2-d worm in a way that can be extended to higher dimensions.

An n-multiworm is an n-dimensional worm, where each dimension is of size n, and each cell contains n. I enter a (3-multiworm)-multiworm.

I have no idea what scale this is.