Solving language for computers seems much like climbing a series of mountains, where each time you surmount one, you realize the next is even higher :) Thanks to deep learning, machines have made rapid gains in speech recognition, as well as improving semantic mapping (a la word2vec and other word embedding approaches).
But once you have a system with human-level speech recognition and semantic mapping, where do we go? The ability to have a meaningful dialogue with a machine seems very difficult to model as a machine learning problem (what constitutes ground truth? What's the reward function look like?), and also has to deal with many unknowns. For example, ask a smart assistant like Alexa or Siri about functionality it wasn't programmed with, and you get a terse "Sorry, can't help you with that." But ask a child, and you prompt a question-answer dialogue (i.e. learning) or perhaps feigned understanding. My toddler son is an expert at giving me the answer he thinks I want to hear, even when he has no idea what I'm talking about :) There are certainly many new problems which we can begin to think about tackling, but certainly no sign IMO that we're running out of applications for deep learning in the field of language.