How to calculate exp(x) for really big integers in Python?
@Paul already gave you the answer for computational question
However - from neural network point of view your problem is indication that you are doing something wrong. There is no reasonable use of neural networks, where you have to compute such number. You seem to forget about at least one of:

Input data scaling/normalization/standarization
small weights bounds initizlization
regularization term which keeps weights small when the size of network grows

all these elements are basic and crucial parts of working with neural networks. I recommend to have a look at Neural Networks and Learning Machines by Haykin.
I'm using a sigmoid function for my artificial neural network. The value that I'm passing to the function ranges from 10,000 to 300,000. I need a high-precision answer because that would serve as the weights of the connection between the nodes in my artificial neural network. I've tried looking in numpy but no luck. Is there a way to compute the e^(-x)