Is there a disadvantage to letting a model train for a large number of epochs?
If you can be sure that the model is not seeing the same instances repeatedly then there is very good chances that your model is not overfitting and that is precisely what you can measure from your validation set, you should see continuously downward loss which will eventually plateau at the local optimum that has been eventually reached by your model, that is the best possible solution attained given the starting conditions. If your model would be overfitting then you should see the loss function start increasing for the validation.

The best way to know if you are overfitting is to take your model and then apply it to a completely new dataset and measure the performance. If the performance is good, then you are fine. 

p.s. To overcome falling inside a bad optimum you can train multiple models each with different starting conditions for the model parameters and then make an ensemble classifier.