Up-arrow notation

Up-arrow notation is a style of notating large numbers invented by Don Knuth in 1976. Its notable use was in the calculation of Graham's number.

How It Works
Addition is an operation where at least two values are combined to produce a new value, known as a sum or summation. For example:
 * $$3+8=11$$
 * $$2+7+16=25$$

Multiplication is an operation where one value is added to itself a certain number of times to produce a new value, known as a product. For example:
 * $$2*8=2+2+2+2+2+2+2+2=16$$

Here, there are eight 2's added together.

Exponentiation is an operation where one value is multiplied by itself a certain number of times to produce a new value, still known as a product. For example:
 * $$3^{4}=3*3*3*3=81$$

Here, there are four 3's.

Likewise, a single arrow in up-arrow notation is the same as taking an exponent. For example:
 * $$3 \uparrow 3=3^{3}=3*3*3=27$$

Two arrows works as follows:
 * $$3 \uparrow\uparrow 3=3 \uparrow (3 \uparrow 3)=3^{3^{3}}=3^{27}=7,625,597,484,987$$

Three arrows works as follows
 * $$3 \uparrow\uparrow\uparrow 3=3 \uparrow\uparrow 3 \uparrow\uparrow 3=3 \uparrow\uparrow 7,625,597,484,987$$
 * $$3 \uparrow\uparrow 7,625,597,484,987=3 \uparrow 3 \uparrow 3 \uparrow ... 3 \uparrow 3$$

From this, we produce an exponential tower of 7,625,597,484,987 3's. This produces a number that is about 3.6 × 1012 digits long.

In general, up-arrow notation works as follows:
 * $$a \uparrow^{n} b=a \uparrow^{n-1} a \uparrow^{n-1} a \uparrow^{n-1} ...a \uparrow^{n-1} a \iff n \geq 2$$

In each instance, there are b a ' s.