Tweer's Number (Size: Massive)

Tweer's number is a massive number which is represented by the Greek letter upsilon.

Representation
$$\upsilon$$

How it's calculated
It all begins with nesting functions.

first we define a function $$f_0$$ being an index into a nesting function (in this case the addition, multiplication, exponentiation chain):

$$f_0(0) = 0+0 = 0$$

$$f_0(1) = 1 \times 1 = 1$$

$$f_0(2) = 2^2 = 4$$

and so on.

Then define f1 as a nesting function of $$f_0$$:

$$f_1(1) = f_0(1) = 1 \times 1 = 1$$

$$f_1(2) = f_0(f_0(2)) = f_0(2^2) = f_0(4) = 4 \uparrow\uparrow\uparrow 4 = 4 \uparrow\uparrow4\uparrow\uparrow4\uparrow\uparrow 4 = 4 \uparrow\uparrow4\uparrow\uparrow 4^{4^{4^{4}}} = 4 \uparrow\uparrow4\uparrow\uparrow 10^{10^{154}}...$$

Now you have a function that can output stupid big numbers with tiny inputs.

But we're not done.

We can define $$f_2$$ as a nesting function of $$f_1$$ and $$f_3$$ as a nesting function of $$f_2$$ as do this over and over again up to infinity.

And we're still not done.

We can define $$g_0$$ as an index into the f functions:

$$g_0(1) = f_1(1)$$

$$g_0(2) = f_2(2)$$

Now we can do the same thing with the g functions as we did with the f functions.

And that's not all.

We can create another series h that does exactly the same thing to the g functions. And we can repeat this an infinite amount of times.

And we can index into the actual tiers of this sequence.

Let's stop here (we could keep nesting and indexing but for simplicity's sake let's stop here). Let's define as an index into this system:

$$TW(1) = f_0(1)$$

$$TW(2) = g_0(2)$$

$$TW(3) = h_0(3)$$

Now we can finally calculate Tweers (Massive) Number

$$TW(\text{Graham's number})$$

Tweer's function can be approximated to $$f_{\omega^2}(n)$$in the Fast Growing Hierarchy, and Tweer's number is $$f_{\omega^2}(f_{\omega+1}(64))$$ in the Fast Growing Hierarchy (written as $$\{G,G,G,G\}$$in BEAF, where G is Graham's Number)