Tetration

Tetration, also known as hyper4, superpower, superdegree or powerlog, can refer to one of two different dyadic operators. The first and most common, written $$^yx$$ or x④y, is defined as $$^yx = \underbrace{x^{x^{x^{.^{.^.}}}}}_y$$, in analogy to $$x \cdot y = \underbrace{x + x + \ldots + x + x}_y$$ and $$x^y = \underbrace{x \cdot x \cdot \ldots \cdot x \cdot x}_y$$. It can also be written $$x\uparrow\uparrow y$$ (using Chained Arrow Notation) or $$\{x,y,2\}$$ (using BEAF.)

The second, less mathematically interesting version (also known as the hyper4 operator) is written x④y and is defined as x④y $$= \underbrace{\left(\left(\left(x^x\right)^x\right)^x\right)\ldots}_y$$, which simplifies to $$x^{x^{y - 1}}$$ as $$\left(a^b\right)^c = a^{b \cdot c}$$. Most of the time, the former definition is preferred.

At the current time, mathematicians have not agreed on the function's behavior on $$^yx$$ where y is not an integer.

$$2\uparrow\uparrow 2$$ = 4

$$3\uparrow\uparrow 2$$ = 27

$$4\uparrow\uparrow 2$$ = 256

$$5\uparrow\uparrow 2$$ = 3,125

$$6\uparrow\uparrow 2$$ = 46,656

$$7\uparrow\uparrow 2$$ = 823,543

$$8\uparrow\uparrow 2$$ = 16,777,216

$$9\uparrow\uparrow 2$$ = 387,020,489

$$10\uparrow\uparrow 2$$ = 10,000,000,000

$$2\uparrow\uparrow 3$$ = 16

$$3\uparrow\uparrow 3$$ = 7,625,597,484,987

$$4\uparrow\uparrow 3$$ = 13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,096

$$5\uparrow\uparrow 3$$ = 1.911012597945477520356404559704x102,184

$$6\uparrow\uparrow 3$$ = 2.6591197721532267796824894043879x1036,304

$$7\uparrow\uparrow 3$$ = 3.7598235267837885389221309308958x10695,974

$$8\uparrow\uparrow 3$$ = 6.0145207536513920379045068488759x1015,151,335

$$9\uparrow\uparrow 3$$ = 4.2812477317574704803698711592765x10369,693,099

$$10\uparrow\uparrow 3$$ = 1010,000,000,000

$$2\uparrow\uparrow 4$$ = 65,536

$$3\uparrow\uparrow 4$$ = 1.2580142906274913178603906646137x103,638,334,640,024

$$4\uparrow\uparrow 4$$ = 2.3610226714597313206877027497781x10 8,072,304,726,028,225,379,382,630,397,085,399,030,071,367,921,738,743,031,867,082,828,418,414,481,568,309,149,198,911,814,701,229,483,451,981,557,574,771,156,496,457,238,535,299,087,481,244,990,261,351,116 = 2.3610226714597313206877027497781x10 8.0723047260282253793826303970854x10153, well over a googolplex

$$10\uparrow\uparrow 4$$ = 1010 10,000,000,000

$$2\uparrow\uparrow 5$$ = 2.0035299304068464649790723515602x1019,728

$$3\uparrow\uparrow 5$$ = 106.00225356799454734996618170266x10 3,638,334,640,023

$$10\uparrow\uparrow 10$$ = 1010 10 10 10 10 10 10 10 10       = Decker

$$10\uparrow\uparrow 100$$ = 1010 10 10 :: :: :: 100 10's :: :: :: :: 10          = giggol

Wow these numbers are so large that you probably can't write its full decimal expansion!