Any friends you make will only become grave resters eventually, and you must steel your heart against love because, for you, love is fleeting. Perhaps you'll never have a permanent home because you have to keep on moving from location to new location every few years before your neighbors (and possibly the government, if you live in a sufficiently advanced society) get suspicious that there's something just not right about you. Some studies have also suggested that you literally experience time more quickly as you get older. Remember when you were a little kid, and thirty minutes seemed like forever? That was because that was ultimately thirty minutes out of seven years. Now, try that when you're nine thousand.
In short, any kind of companionship from mortals will only last (from your perspective) an instantnote .
Even if it's possible for you to surround yourself with immortal friends, there are only so many places to go and life-defying antics you can perform (and tropes you can browse) before life settles into a repetitive boring monotony.
Eventually you'll even see your civilization (maybe even species) die out and make you a Living Relic. And then there's the fact that the memories of things that the world will never see again, one of the few rewards of immortality, may also one day vanish from your mind by sheer dint of memory capacity. Senility will be a blessing because you won't remember that you've done the same thing ten million times.
Meanwhile, the sun will slowly brighten as it ages. In a little over a billion years, it will have brightened enough that liquid water will no longer be able to exist on the surface of the Earth. Maybe you can move to Mars before this happens. But five billion years from now, the sun will run out of fuel in its core entirely and swell up to red giant size, swallowing the Earth and making Mars too warm to live on. Move to the moons of Jupiter? That'll only work for a couple million years tops; the sun's red giant phase will then end, and it will shrink to a tiny, cold white dwarf. You'd have to find a new solar system to move to, if you wanted any hope of living on a habitable planet.
And after some more billions of years, that new star system you've moved to will suffer a similar fate to ours. You'll have to pick a new solar system again. And again. If you're lucky, the cosmologists who predict the Big Rip will be right, and in a paltry 20 billion years the universe will be expanding so rapidly that your atoms won't even be able to hang together any more and your existence will finally end. If you're lucky. But if the cosmologists are wrong about the Big Rip, then after a while of hopping from one star system to another, you'll be faced with a new problem: as the stars die, fewer and fewer new stars are born to replace them. After a few hundred billion years, the only stars left in the galaxy — in the whole universe, in fact — will be the long-lived red dwarfs, each so dim that the range of distances from them inside which a habitable world can exist is horrifically slim. You'll have to search a lot of red dwarfs before you can find one with a planet at the right distance, and there'll be no guarantee it'll actually be inhabited. It will become harder and harder for you to find a planet to call home.
Eventually, say in a few trillion years from now, the Universe itself shows its mortality (lucky bugger) and dies. As you hop from one dying planet to another, if you can, eventually they too will end. Soon, it'll just be the endless void of subatomic particles that can never reform into anything. And you, floating there. Forever. Unless, of course, you get lucky and it turns out all matter in the universe will contract into one small mass and then make a new Big Bang, in which case after a sufficient amount of time (say, a few billion years) you might be able to find life to talk to. Assuming you survive the Big Crunch, of course... Then again, such life would likely consider you an Eldritch Abomination.
Also consider this, since you cannot be hurt and therefore are bereft of mutability, your entire species will probably evolve into something else. Imagine how you would seem to humans (or their descendants) in ten thousand years, or a hundred, or a million. You would be as simple and primitive to them as a neanderthal would be to us. There is a hope spot though. Provided a civilization lasts that long, they could show you mercy and maybe reverse the effect, or could develop technologies that would enable them (and you) to escape the eventually dying universe altogether, though this would probably take a very, very long time.
Time Abyss doesn't even begin to cover it.
Scared straight yet? But we didn't even get started on the technical details. Maybe you were fortunate enough to start your eternal life at a time when you were grown up and healthy. Then again, if you got "blessed" with unaging immortality before you fully grew up, you Can't Grow Up. Ever. Or, if you have acquired it at an old age, you are to stay a fragile old man forever. And that's assuming that you're always staying the same age was part of the package. If you've got Age Without Youth, you'll be lucky if you become only a grasshopper after your wrinkles and bent back take over.
And what about injuries? Did you remember to ask for a Healing Factor? Good luck to you if you didn't, because now you are at a daily risk of obtaining some really crippling injuries that will stay with you forever. Count yourself lucky if it ends with only going comatose, because you may very well get crippled to a degree where you cannot even move, cannot even speak, but still going on, forever and ever...
Or maybe you will fall into a deep pit or sink into the ocean, or get thrown into a Torture Cellar from whence you can not escape. Or worse, Room 101. A different person would starve and die, but you will have to wait there often for centuries — and that's even assuming that someone will eventually find you. No wonder so many immortals end up crazy...
Of course, many of the above points assume that you end up invulnerable and "truly" immortal. A person who has given this a bit more thought can opt/ask/strive to simply remove ageing and indeed, that version would be mercifully bereft of many of the above points (due to the possibility of dying and the ability to change one's body to suit one's preferences or needs).
There's also the issue that being immortal doesn't automatically mean having perfect health. What happens if you get sick? Sure, your immortal immune system would probably fight off in no time, but imagine how many airborne, waterborne, and sexually-transmitted diseases you would inevitably come in contact with, complete with all their nasty symptoms. Even if you did build an immunity to all of them, you could still become a carrier. Imagine getting an STD once and never being able to have safe sex in your immortal life again. Furthermore, could the diseases end up acquiring your immortality as a result of infecting you? You could become the carrier of the most dangerous disease in history without even realizing it.
There are also those chronic diseases and the ones that do horrible things to your body, like AIDS, Leprosy, Foot and Mouth disease, and Necrotizing Fasciitis. Imagine being stuck with one or all of those forever.
And what about cancer? Since tumors are the result of mutations in your own cells, they could be immortal as well. You would be stuck in an eternal battle with the painful, bloody, debilitating symptoms as the tumors metastasize all over your body with nothing to stop them. Maybe your whole body would eventually become one big semi-humanoid tumor.
Maybe you're thinking that you could be relatively lucky and become one of The Needless. You won't have to worry about eating, drinking, sleeping, or even breathing ever again. Except, without using them or needing to use them, your organs will gradually shut down, meaning Eating Optional no longer applies. Your sense of taste will disappear, along with any joy that you may have gotten from eating your favorite foods. No sleeping also means you'll find yourself with even more time that you'll need to spend looking for something to alleviate your boredom.
Speaking of boredom, that may be your ultimate adversary. Really think about how long it would take you to do all the things you've ever wanted to. Every game you've ever wanted to play, every subject you've ever wanted to learn, every book to read, every show to watch, every song to listen to, every place to visit, every puzzle to solve, etc. All of that might take you a few hundred or a few thousand years to accomplish. Then what? After living for millions or billions of years, you're going to run out of things to do eventually. Things are going to start feeling repetitive, tedious, or just not as fulfilling as they used to, especially not if your immortality comes with a perfect memory.
And that boredom would equally apply to society. With social and cultural trends that rise and fall, you're bound to see the same ones rise in popularity, fade away, and then come back. It only took a few decades for zombie media to regain mainstream media attention, then fade. Superheroes, westerns, romances, adventure stories, you're going to start seeing the same things over and over again. Typical tropes and cliches may be so ingrained in all of them that you recognize them and it'll make everything feel identical. When you've literally seen it all, what is there left to see?
All of that is assuming you can even keep up with the cultural changes of society. With changing technological and medical advances, can you even comprehend a world that moves forward at an ever-increasing rate? Imagine how much of a culture shock someone from just a hundred years ago would have if they found themselves in the modern world. Can you picture yourself like that? What about a society a thousand years in the future? Ten thousand? A million? Will their forms of entertainment even be comprehensible to you? Will you be able to learn how to communicate with them? Can you even interface with their technology? Sure, maybe you could learn, but no matter how much you do, you'll be right back to square one: how long before you exhaust everything you wanted to do?
And that's in the most optimistic scenario. Worst case, you outlive civilization or humanity itself, and then you can be sure there won't be any new media for you to enjoy. Hope you enjoy listening to all your songs on shuffle for the 4,000th time.
And if you think that returning your immortality can help you with all this, think again.
Any characters who initially jump for joy at the prospect of living forever will find within a few centuries' time that immortality maybe isn't all it's cracked up to be. Their quest will become a search to find a way to return to a normal life. Sometimes they'll even allow themselves to be killed outright by supernatural means to be able to rejoin the natural life cycle at last. Sometimes the ones who can be killed but not age are rather picky; they won't commit direct suicide, they just want to be able to grow old and eventually die of organ failure. Or they slowly forget things over time. (Real-life senile dementia sort of says otherwise, but this is fantasy we're talking about).
One of the biggest reasons Cessation of Existence (formerly a primal fear) is considered to be a comforting heaven by many people nowadays, especially the more atheistic, cynical and existentialist ones, after giving a deep deconstructive thought about immortality and realizing the inherent Fridge Horror within.
Oddly, elves, Gods or naturally immortal species are portrayed to be immune to this (although in some cases they too may grow weary of immortality, given enough time). Maybe they're good at finding hobbies? At the least, living among fellow immortals is a good way to ameliorate the strain. After all, they've had a lot of time to work out cultural mores and general psychological structures necessary to deal with the relatively unchanging social landscape. (They usually have trouble with procreation, though). It's also worth noting that, as an immortal, you have an infinite amount of time to deal with these problems, which may mean simply making peace with them, or even somehow finding a solution that does not necessarily involve ending said immortality.
It's not all hopeless, however. In some cases, you actually are better off with living forever. You may be bored, but at least something is going on. Although, technically that is living for ever too, you just happen to be doing it in the afterlife. So basically the "out" someone suffering from this trope is looking for is Cessation of Existence, since an afterlife of any sort would just give them a new place to explore for a bit before they start to notice all the same problems all over again, unless said place somehow removed the concept/sensation of boredom and saturation.
Also, bear in mind that the reasoning for why immortality would suck is largely based on our present societal, moral, cultural and cognitive views, which needn't be correct or don't necessarily encompass everything there is to be known about everlasting life and/or the finite nature of things (there are things we know, things we know that we don't know and things we don't know that we don't know). Us arguing about such a concept would make as much sense as two hunter-gatherers arguing and debating the merits and pitfalls of modern civilization.
So to wrap up. In order to properly "enjoy" true immortality in a way that does not involve the radical alteration of your being or some sort of deus ex machina, you would have to never start relationships with mortals, have some sort of time and memory control, be able to travel a lot, and tons of other stuff as well.