Move matches and make largest possible number

a)
Is it:

b)
Is it:

a) Not certain this is the biggest, but I can't find anything bigger than

b) Moving 4 matches allows

which is pretty big. Not sure if it's the absolute biggest, but it's up there.

This is, of course, assuming that the digit 1 must be two sticks high. There's definitely a higher ceiling if it can only be one.

111153

Moving 2 Match sticks, the 4s can be made into 11s. And the third match stick would be moved to get 5.

Hard to go wrong with

As obviously

Moving 3 matches, I'd

giving

My best guess for 4 moves is to

to get

I think that's the highest so far, if each one needs two matchsticks.

For a)

For b)

Moving 4 matches to form

         _   _
| | | | |_  |_
| | | |  _|  _|

and viewing it upside down gives S S 1111 where S 1111 is the maximum shifts function of the busy beaver game with 1111 states http://en.wikipedia.org/wiki/Busy_beaver#Maximum_shifts_function_S . S 6 is 3.515E+18267 and S 7 is already 10^10^10^10^18705353.

S S 1111 is the maximum shifts function of the busy beaver game with S 1111 states, a number uncomputably large, but still finite.

(b) Move 4 matches to form

      _      _
| |_| _|  / |_
|   | _| /  |_

The slash has to be squeezed in between the 3's to make 143/ε. The last character represents epsilon, an arbritarily small positive quantity used in the definition of limit in calculus, which is greater than zero to avoid division by zero.

(a) Move 3 matches to form

     _  _     _
|_| | | _| / |_
  |   | _|    _

or 473/ε. The vertical match of ε is centered over the middle bar.

(b) Move 4 matches to form

   _   _ 
| |_| |_|  |  |
|   |   | _| _|

which viewed upside down is Γ Γ 661, where Γ is the gamma function http://en.wikipedia.org/wiki/Gamma_function

Γ 661 ~ 10^1576 and Γ Γ 661 ~ 10^10^1579

which is slightly larger than 7^7^473 ~ 10^10^400.

Interpreting it as Γ Γ bb1 where bb1 in hexadecimal is 2993 in decimal,

Γ 2993 ~ 10^9103 and Γ Γ 2993 is too large to compute in Wolframalpha but is larger than Γ Γ 661.

Even better are

    _   _ 
   |_| |_|  |  |
||   |   | _| _|

Γ Γ 66¹¹ and Γ Γ bb¹¹, respectively.

(b) Move 4 matches to form

            _  _
|_| /| /|  | _|
  |          | _|

or 4 ↑↑ 73, where ↑ is Knuth's up-arrow notation http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4 (73 4's) is too large to compute in Wolframalpha.

            _  _
|_| /| |||  | _|
  |          | _|

A convention allows multiple ↑'s to be specified with a superscript, so 4 ↑¹¹¹ 73 = 4 ↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑ 73 (111 ↑'s), a number too large to represent even with power towers.