Find the largest fragile prime
Python 2 - 126 1221 1337 1719  2268 digits

999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999799999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

'9' * 1944 + '7' + '9' * 323


There are at about len(n)^2 resulting numbers of Remove(n, startIndex, count). I tried to minimize those numbers. If there are lots of digits next to each other are the same, lots of these resulting numbers can be ignored, because they appear multiple times. 

So I took it to the extreme, only 9s and a little prime in the middle. I also took a look at the fragile prime under 1 million, and saw, that there are such fragile prime. Searching for numbers with 2 9s at the end works really good, not sure why. 1 number, 3, or 4 9s at the end results in smaller fragile primes.  

It uses the pyprimes module. I'm not sure, if it is any good. It uses the miller_rabin test, so it's probabilistic. 

The program find this 126-digit fragile prime in about 1 minute, and for the rest of the time it searches without success. 

biggest_found = 80651

n = lambda a,b,c: '9'*a + b + '9'*c

for j in range(1000):
   for digit in '124578':
      for i in range(2000):
         number = int(n(i,digit,j))
         if is_prime(number):
            if (number &gt; biggest_found):
               if all(not is_prime(int(n(i,digit,k))) for k in range(j)):
                  biggest_found = number
                  print(i+j+1, biggest_found)
            break


edit:

Just saw, that you removed the time limit. I will run the program over night, maybe some really big fragile primes appear. 

edit 2:

Made my original program faster, so but still no solution with more than 126 digits. So I jumped on the train and searched for x 9s + 1 digit + y 9s. The advantage is, that you have to check O(n) numbers for primality, if you fixe y. It finds a 1221 rather quickly. 

edit 3:

For the 2268 digit number I use the same program, only divided the work on multiple cores.