The European Court of Justice says that we have a "right to be forgotten". Before we can determine if that claim is correct, we need to understand what the right means.
For our purposes "rights to" come in two different variants. The first is a liberty right. Thus your right to roam is a liberty: you have no duty to anyone else not to roam and are therefore free to do it.
The second is a claim right. Claim rights consist in the duties of others. Thus your right to a living wage consists in the duty of certain others to pay you that wage.
Which form of right is the "right to be forgotten"? Clearly it is not a liberty. The forgetting or remembering is not done by the right holder but by others, and it makes no sense to speak of a "liberty to be forgotten".
If there is a right to be forgotten then it must be a claim right that imposes duties on others. But what duties does the right to be forgotten impose? The duty to forget, of course.
Now the extraordinary nature of this "right" reveals itself. A right to be forgotten would create duties in the most intimate spaces of our mind. It imposes limits on what we are permitted to remember and establishes claims about what we are required to forget. It is an attempt to regulate thought itself. When analyzed clearly, it is easy to see that there can be no such right.
Now of course the European courts do not seek to create legal accountability for thought (not even the totalitarian regimes of the 20th century attempted that). Instead the law requires Google to have a process to remove certain documents from its search results that european citizens would rather have forgotten.
In this sense one might think that the phrase "the right to be forgotten" is a little more than an alarmist misnomer. We should rather talk of a "right to have no longer relevant documents removed from public search".
But the problem cannot be so easily evaded. The more mundane claims regarding documents and search are supposed to be explained and supported by the deeper and more intuitive right. Why should Google remove documents that violate no laws against liable, defamation or hate speech? Because there is a right to be forgotten!
But there is no right to be forgotten. There is not even a right to be remembered fairly. This is for the simple reason that thought is free. We reject the notion that what we remember, and what we forget, can ever be subject to moral or legal constraint.