A neural network that keeps learning as new training data becomes available
You can achieve that by training the network using batches of 100 examples. In this case, you train on a batch and update the neural network parameters, you predict on the second batch to estimate the accuracy and then you use the examples of this second batch to continue training the network and so on and so forth.

Batch training (a form of online learning) is implemented on every framework, see for instances here for Keras.