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Actually, this does not work on sage-5.9 either. apeel seems to use a 32-bit system whereas ppurka seems to use a 64-bit system. On a 64-bit system, you will only be able to go twice further (65536 = 2^16 instead of 32768 = 2^15):

sage: f = x[0]**(2^16-1)   
sage: f = x[0]**(2^16)  
OverflowError: Exponent overflow (65536).

If you really need to work with such a big exponent, you could request it on the trac server and then on singular as well, though it is possible they won't change this limit due to a possible loss of performance (i didn't check the code, so it might be possible to go until 2^32 without much loss).

If you are looking for a mathematical workaround, you need to give more information on what is your function doing. For example, if you only use the variable x[0] in this function, you could try to transform your polynomial f as a univariate polynomial, and then work with it:

sage: P = f.univariate_polynomial()