mathematics - Largest number with five 1's and five numeric operations

You have five 1's at your disposal, together with five arithmetic operations of your choice. However, as you only have five operations, you should choose them wisely.

Question: What is the largest integer that you can generate this way?

Rules:

• Numbers can not be infinite. No dividing by 0.


• You cannot concatenate the 1's (i.e. you cannot use two 1's to make 11)


• You cannot use any other numbers in any other form: no Greek alternatives, no constants such as $e$ or $\pi$.


• Parentheses come for free; you may use as many as you like.


• You may use two or more operations in a row


• You may use any notation you would like. One solution below uses "Knuth's Up Arrow Notation". Each arrow uses one operation of the five allowed operations.

Examples:

  1+1+1+1++1 = 5

  ((1+1+1)↑↑(1+1)) = 27  <-- Uses Knuth's Up Arrow Notation

  (1+1)^((1+1+1)!) = 64

  ((1+1+1)!)^(1+1) = 81

I have posted my solution below, let's see if you can beat me!

Answer

You may use any notation you would like.

Browsing Wikipedia I found the Steinhaus–Moser notation.

If all ones must be used we can start with:

(1+1+1)^(1+1)=9

With one operator left we can put the number in a circle:

⑨

According to Wikipedia already ② is too big to be displayed. If less than 5 ones can be used, we can make the number even larger.

Update:

Instead of a circle we can use any n-sided polygon to make the number arbitrarily large. See for example the definition of Moser's number in the article linked above.