There are a couple things that may have been overlooked so far.
- It is possible to reverse the aging process (so you can go back to the physical age that you like most)
But if everyone after the initial crop of adopters is becoming immortal at, say, ages 20-35 (depending on children if there is a sterilization policy), that radically decreases the range of physical ages available to them. Is physical age-changing a one-way procedure or can you increase your physical age as well? Think about all the child vampire stories you've read. Think about all the child soldiers that actually exist in the world today--if the immortality procedure could be induced involuntarily, there are plenty of corrupt governments and insurgent groups/freedom fighters that would jump at the chance for eternally childlike armies or fleets of childlike secret agents, or master thieves, or whatever. They would be young, limber, and small forever, which is a huge training window for those precise tasks for which children's physical structures are ideal. And if the immortality 'bots can replicate on their own, you would only need to raise the funds for one treatment and then systematically spread them throughout your chosen population. Etc.
- You can die by accidents or diseases that are normal for your age
But in scenarios that have been discussed already, what is considered "normal" in a world where tech immortality is not yet possible is not an accurate gauge for what will be "normal" after a couple of years of availability and adoption. Depending on whether the nanomachines could communicate wirelessly (but this would still work if they couple together or send bloodstream packets or what-have-you), someone could set up a virus that would destroy the nanobots or give them faulty instructions to attack their host--a problem that is definitely not normal for any age at the moment.
- You won't age (and therefore not die of age)
What exactly is meant by "dying of age"? Isn't "dying of old age" not of a disease common in the elderly currently a function of cells not being able to repair themselves and organs wearing out and so forth? If that problem is solved by some technical procedure, cf my response to the previous point, in that in a world where cells and organs are more robust it is conceivable that, even if no specific advancements were made toward curing or vaccinating against things like cancer and infectious diseases, "normal" diseases for an age group could also be rendered ineffective. Until a mutated strain successfully infected the world, of course, depending on the tech solution for the immortality...