What is the largest number you can create with three digits, or three one-digit numbers?

Question

I know most people think the answer is 999, or 9^(9^9), but there are numbers larger than that, without using factorials. Can you come up with a few? So, from now on, no factorials.

Oh, and no infinite unary operations. You can use factorials if you like, or any other unary operations, but it can be only one per number, otherwise you will have to use a 9 or a bunch of 9's to identify how many operators there are.

Answer 1

A few notes to start off:
9^9=9^2•9^7=81•9^7

Therefore 9^(9^9) >>9^99=9^(9•11) =(9^9)^11>(9^9)^9>> 99^9

Thus the largest number using only 3 9s would be 9^(9^9) (this can be written without any mathematical symbols).

Using Knuth's up-arrow notation, you can write something much larger:
if you were to allow one mathematical operator, then you could write 9↑(9^9)=9^(9^9)

If you used n mathematical operators, then you could write:
9↑↑↑ . . . ↑↑↑9^9 (where there are n ↑s)

But you can actually do something a lot more efficient. If you use only two mathematical operators, then you could switch to Conway chained arrow notation. Using a three chain, you could write
9→9→9=9↑↑↑↑↑↑↑↑↑9= 9↑↑↑↑↑↑↑↑(9↑↑↑↑↑↑↑↑9)

This is a number so large that if I made a tower of 9s (9^(9^(9^(9^(9 . . .))))), there is not enough space to write all of the 9s here. This number is absolutely huge.

But there is something even more insane:

Steinhaus-Moser notation:
If you take the number 9^(9^9) and place it in a triangle, this number is represents (9^(9^9))^(9^(9^9)) Very large number, although not as large as 9→9→9, but only uses one mathematical symbol. . . and we can get even larger:

Take 9^(9^9) and place it in a square:
This represents 9^(9^9) in a triangle, that in a triangle, that in a triangle . . . until you have 9^(9^9) triangles. Ok, now we have an insanely large number, but we can always go larger.

The largest number (to my knowledge) written with only 3 9s and one mathematical symbol is 9^(9^9) in a circle. This is equivalent to 9^(9^9) nested in 9^(9^9) squares.

It has been proven that 10 in a circle is still smaller than Graham's number (the largest (known usuable) number), but I have a feeling that 9^(9^9) in a circle would be larger, not really sure though. But in any case, the largest number that you can have with 3 9's and n mathematical symbols (to the best of my knowledge) would be 9^(9^9) nested in n circles. The possibilities are endless.

The explanations of Knuth's up arrow notation, can be found in the link below.


And if you want to be anal about everything, you can use some other base. For instance, you can assign all 127 ASCII characters a value and work in base 127. Therefore you can use 127 instead of 9. You can actually work in any base you want to (go as high as you want to) and use a "digit" that is as large as you want it.

Answer 2

999

Answer 3

999

Answer 4

9*9*9

Answer 5

9!!!!!!!.... - infinite factorials

or if factorial is not allowed, 9^(9^9)
(which is larger than (9^9)^9 )

Answer 6

Well considering in Roman Numerals characters are actually the digits...
Hence, 3 digit MMM in Roman has value of 3000.
So I say that would be the largest number.
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oooo, but I like Eulercrosser's answer. well most importantly i like the source of that answer.

Answer 7

9^9^9=1.966270505...x 10^77

Answer 8

I would go for the obvious, 99 / 0 = infinity!

Answer 9

9!^9!^9!

Nine exponent to the power of nine exponent to the power of nine exponent

(81^81^81)

Answer 10

If we can't use factorials, can you tell us which mathematical operations ARE allowed?