Jack's a Celebrity.

One of the things which makes Ma and Jack's lives harder in Room after their escape is the fact that they have become famous, with the media trying to get people interested in their story by twisting public perception of their time in Room, and ordinary people becoming fascinated with them, especially Jack, and their story. Much of this fame is portrayed as a hindrance to Ma's and Jack's adjustments to normal life, as they try to hide from cameras and keep out of the public eye. Jack, on the other hand, appears unphased by the attention he gets.

There are several instances where we see Jack interacting with his fame without a full understanding of what's happening: Ma carries him past the "vultures" into the police station, he sees himself and Ma on TV, he signs autographs for people at the mall, etc. In none of these instances do we see him opposed to his stardom, which one could argue is because of his lack of familiarity with the concept of fame and what it may mean for his life. However, I would argue the opposite: It doesn't take Jack long to comprehend what his fame means, and, though he clearly does not grasp the effects it could have, he understands it clearly enough to enjoy it.

The first time Jack has the opportunity to recognize his new fame is when he watches the television in the doctor's office right after the escape. He sees himself and Ma on TV, and it greatly intrigues him. In this instance, Jack's fascination is likely a result of his inaccurate understanding of television, but he was still quite interested in seeing himself to the point that he became somewhat frustrated that the receptionist turned the television off. This is where we first see the trend which persists throughout the novel of other people becoming extremely concerned about Jack's fame and his own recognition of it while he only finds it enjoyable.

During this first encounter, Jack likely doesn't realize that there are thousands of other people watching him on the television as well, so his lack of concern does not yet reflect an affinity for his fame. However, subsequent encounters indicate that once he understands that people recognize him, his appearance, and his unique experiences, he has no problem with it, even attempting to take advantage of his popularity. This can be observed as he is taken out with Deanna and Paul, and he intentionally reveals his identity despite having been told that it was important conceal it. In this case, he uses his fame to spite Paul and Deanna and get the book he wants. It works extremely well, as he gets what he wants and appears not to suffer any negative effects, which is likely what he wished and expected.

This shows Jack's ability to recognize his fame and use it to his advantage, but the incident which proves his affinity for his fame and the public's recognition of him is the incident at the mall with his grandmother. For the few moments during which she is separated form Jack, he is discovered by people who figure out who he is, and he willingly signs autographs. Jack's narration of this scene includes mostly direct quotes, indicating that he is not totally sure of what his fans are thinking or why they're so interested in him, but he notes that they were his new friends, which, along with other instances with strangers, indicates that he enjoys socializing with new people. Since his stardom helps him meet new people and create bonds he considers friendships, from his limited perspective, public recognition of Jack can only be to his benefit.

From a broader perspective, one could argue that Jack's fame is problematic for his development, but I would again argue the opposite: his fame is neither beneficial nor problematic. As Jack must adjust to an entirely new world with oodles of new concepts and norms to learn, learning to deal with fame is not as shocking as it would be to a normal child. It's only another thing to which he must adjust, and to him, it likely doesn't stand out as something any stranger than the rest of the things he doesn't understand. In light of the numerous other things facing Jack, fame is only a small hindrance which he won't have additional trouble adjusting to. Therefore, the fact that he enjoys his fame in the process of learning to navigate it is as beneficial to Jack as he believes it is.