This answer depends very much so on the type of neural network and algorithm used for training.
If you are using gradient descent on a neural network of one input layer, one output layer, and no hidden layers there are many functions that you can't learn. One simple one is the XOR function. Due to the fact that XOR is not linearly separable, it can not be represented by a neural network with no hidden layers.
If you are using NEAT to build recurrent neural networks then all functions(**) can be represented given enough time and data. This is due in part to the fact that recurrent neural networks are Turing Complete.
One of the biggest causes for limitations when using neural networks is based on the difficulty of interpretation as to what the network is doing. The network is gradually building up an understanding of the function as it goes from the input layer to the output layer, but it is very difficult for us to understand this building up process and interpret what the neural network is attempting to do. This makes it very hard if not impossible to manually tweak your neural network in a meaningful way.
Another limitation is the need for training (in large amounts) in order to have a meaningful representation of your data. Neural networks have a tendency to need large amounts of data before converging to a meaningful hypothesis space. This has resulted in clever algorithms to generate training data without needing human interaction, such as Generative Adversarial Networks, but the underlying problem remains.
** Not all functions can be computed by neural networks, however all computable functions can be. An example of an uncomputable function is the mapping of all programs from the program to whether or not this program will halt (The Halting Problem).