I don't see why anybody could possibly believe that immortality has a downside of any kind. The only issue I'd have was if I was the only person in my circle of friends that got it, and then had to watch them waste away with age while I remained the same. The idea of being bored as an immortal is totally absurd. How long would it take you to read all the books in the world? How long would it take you to watch as the television shows and movies in the world? It is literally impossible to keep up to date with all the new creative content the world produces on a daily basis, so the idea of getting 'bored' as an immortal is quite laughable.
Another factor I think people tend to overlook when they consider the consequences of biological immortality is the lack of urgency that suddenly sets in. Just about every long-term decision people make today is based on the idea that they have a limited amount of time to accomplish their goals in. People are pressured into going straight to college and then university because they'll only have around 20 to 30 years worth of productive time to truly make a career when they get out. If you're immortal, you don't have those constraints. You can take a minimum-wage job and then spend the majority of your free time surfing the internet or reading books at your local library forever. You'll never become unemployable due to health concerns or aging, so your incentives to improve your own education and take on more responsibilities are now motivated solely by your desire for self-improvement.
Furthermore, if you were immortal, then the quality of your life would only improve as time went by - theoretically, at least. The more experience you built up, the more capable you would be of handling new challenges and new fields of learning. Say you spent 300 years studying languages; how fast would you be learning new languages at the 250 mark compared to how fast you started learning them right at the beginning? It would be the same with science, biology, mathematics, etc. Even better, as you became more and more adept, you'd be able to contribute as well as learn from others. You could apply this to any field of learning, with no limitations. If you kept your resume updated and had a good reputation, you could walk into and out of careers at the drop of a hat, with your expertise alone guaranteeing you a position on merit wherever you liked.
These are just a few points I thought might be relevant to this discussion. It's like 7AM, so hopefully this doesn't come across as incoherent.