Which word describes a person who exaggerates their skills to gain approval?

The term that best describes a person who exaggerates their skills to gain approval is "sycophant." This word refers to someone who acts obsequiously toward someone important in order to gain advantage or favor. Sycophants often inflate their own abilities or the qualities of others to gain admiration or approval from authority figures or those perceived as powerful. This behavior is typically self-serving, as the sycophant seeks to establish a favorable relationship or benefit personally.

In this context, the other options do not quite capture the essence of exaggerating skills for approval. A philosopher engages in the study of fundamental questions about existence and knowledge, without the intent of exaggeration. An idler refers to someone who is inactive or lazy, exhibiting no particular ambition to boast about abilities. A pretender, while someone who may pretend to have certain qualities or skills, does not necessarily do so explicitly for the purpose of gaining approval, which is a key aspect of being a sycophant. Thus, sycophant aligns perfectly with the behavior described in the question.