Word2vec only have one hidden layer followed by a softmax layer. If we add more hidden layer(fully connected feed forward layers), then the model complexity is increased and likely we will get a more powerful model. As far as I know, DAN(deep averaging network) uses 2 feed forward layers. But why we haven't seen such models in word2vec published?
Probably because the model becomes more complicated than necessary. While deep networks can accommodate more non-linearity which makes them more powerful in certain tasks, adding more non-linearity than necessary will increase training cost with no significant improvement in performance. Deep Averaging Networks (DANs) are used to obtain sentence/document embeddings, which is a much more complex task than obtaining word embeddings, so adding more hidden layers is required. Recent models focus on using the input in more intelligent ways, such as using attention. They usually use 1-2 hidden layers.
Also see this question can we make a word2vec nn of more than 3 layers using tensorflow.