Its a bit messy but alright:

 

Basically, the problem is to find the remainder when the number is divided by 100

 

(14)^{14^14} = (196)^{7^14} = (100n-4)^{7^14}. = 100n - 4^{7^14} (not the same n)

 

So, now we have to find the remainder when 4^{7^14} is divided by 100.

 

4⁷ = 16384 = 16400-16 = 100n-16

Hence the remainder when 4^{7^14} is divided by 100 is the same as when 16¹⁴ = 4²⁸ is divided by 100. This may look intimidating but it can be resolved quite simply.

4⁶ = 4096 = 51*100-4 = 100n-4.

4²⁸ = (4⁶)⁴ * 64

     = (100n-4)⁴ * 64

     = (100n+4⁴)*64

    = 100n+4⁸

    = 100n + (100k-4)*16

    = 100k-64

 

Summarizing, 14^{14^14} = 100n-4^{7^14}

                                = 100n-(100k-64)

                               = 100m+64

Hence the last two digits are 64

 

I said it would be messy