Busy Beaver is a good start, and they got to it within about 15
minutes of the start of the contest. Chaining busy beavers will also
get you much larger, but by that point, someone had already chained
exponentials in a similar fashion, so it wouldn't count as a new
trick.

The next trick was to define the second-order Busy Beaver function,
BB_2. I'll assume you're familiar with the definition of the Busy
Beaver function; If not, I suggest Wikipedia :). BB_2 is the busy
beaver function for Turing machines that have the ability to solve the
halting problem on normal Turing machines (e.g. suppose they have a
magical oracle attached that knows the answer to the halting problem
for any machine). Such a machine can therefore compute Busy Beaver in
a small number of steps, and so can produce numbers much larger than
the standard Busy Beaver function.

So, the next entry was BB_2(10^100)

Of course, we can continue: Define BB_3 to be the busy beaver function
for machines with halting-problem oracles for second-order turing
machines, BB_4 to be ... and so on.

But, before we get caught up in games trying to increase the
subscript, we can go even further. We can in fact define BB_aleph-0,
to be the busy beaver function for a Turing machine that has a oracle
that answers the halting problem for *any* finite-order TM. Note that
even though this definition now incorporates an infinite quantity, the
number defined is still finite, since it is defined in terms of
finite, halting, turing machines, along with oracles that only produce
finite numbers.

So, next entry is BB_\aleph-0(10^100)

But, why stop at the *first* infinite ordinal? We can define
BB_(\aleph-0+1)(10^100) as the busy beaver function for turing
machines that have the same oracle as above, as well as being able to
solve the halting problem for order-aleph-0 TMs. And, in fact, we can
define a Turing machine for any ordinal (any size of infinity), with a
corresponding Busy Beaver function.

So, the next entry was BB_(\omega^{\omega^{\omega^{\omega}}}(10^100))
(\omega means the same thing as aleph-0, when we're discussing
ordinals).

At this point, the contestants had pretty much exhausted Busy
Beaver. You can, of course, keep nesting these functions, and keep
putting in larger infinities, but neither of those is fundamentally
new. However, there was one *big* trick left before the contest ended.

Every concept mentioned so far, can be rigorously defined in terms of
mathematical set theory (I'm not sure if you're familiar with this,
but mathematicians have basically settled on pure set theory as the
basis of all mathematics, and everything else is, in principle,
defined in terms of sets). Therefore, conceptually, we could define a
larger number than *any* of these by saying something like:

``The smallest number larger than any representable in standard set
theory using fewer than 10^100 symbols''

However, this doesn't quite work, because we don't have a rigorous way
to define ``representable''. It turns out that doing so would be
equivalent to having a truth predicate for set theory -- that is, a
predicate, in set theory, that when applied to a (encoded) statement
of set theory, tells whether or not that statement is true. And, in
the 1930s, Alfred Tarski proved that no consistent axiomatic schema
can contain its own truth predicate.

However, let us define a second-order set theory. In standard set
theory, you can take some expression, e.g.

Runs(Bob)

and replace some object with a variable, and quantify over it:

\exists x : Runs(x)

In our second-order set theory, you can quantify over *predicates* as
well, e.g.

\exists x : x(Bob)

Now, it turns out, this set theory is completely well-defined, and,
furthermore, it is powerful enough to contain standard set theory's
truth predicate. Therefore, in this *second-order set theory*, the
above statement is well-defined, and we can submit that definition as
our number.

After the MIT professor finished this explanation, the Princeton
challenger literally fell over onto the floor, and the judge declared
the MIT professor the winner.