I am experiencing swelling on the tip of my toe (2nd toe). there's no pain or bruising, but the swelling continues after 3 weeks. ?

Swelling: You should be evaluated ,an X-ray would be of benefit as a start Podiatrist evaluation

Varies: There are many potential causes, I would not hesitate to be seen. Some people will have a "hammer toe" particularly common at 2nd toe and a callus or sore can develop at the tip of it. There are other potential causes including inflammatory and infectious causes as well. Can exam would likely reveal the most likely diagnosis.

Longer second toe?: There is a host of reasons why the tip of the second toe is swelling. Swelling is more common when the second toe is longer than either the great toe or the third. The retrograde forces from the wearing of socks or the use of shoes with elevated heels can cause a low grade blunt force trauma to the tissues on the tip of the toe. Toe remove all doubt have your toe evaluated by a specialist.