How create an AI that continuously adapts to different users? I want to make an AI with deep learning which can adapt itself from user to user. 

Let's say we have food combiner AI which suggests a food to eat with another food as you give as input. This is the most personalized case I found to ask here. For example the AI suggested a food for me. However, the food AI suggested for me might not be good choice for another person. So another person will let the AI know like "I don't like that food to eat with this. Etc. When the user let the AI know that, It should affect AI's further combination food suggestions. 

How can I build that AI? Where should I start from? which area or topics should I research?
Assuming you have enough training data and representational capacity, you can give each user a unique identifier and concatenate that with the other inputs to the neural net. As users give more feedback, the network will learn in the usual way since each situation (e.g. food-user pair) is represented by a unique input.

I'd consider this a general, brute-force approach, and it may not be suited to your application. Others' links to recommendation systems might be more useful. It depends on what your task is and what your constraints are.