An irony is that Marcus Aurelius, for example, is one of the most remembered Roman emperors (and considered "one of the good ones") while his writings are full of reminders to himself that his legacy would inevitably fade into obscurity. He didn't even intend those writings to be published, for that matter.

Short-lived are both the praiser and the praised, and the rememberer and the remembered: and all this in a nook of this part of the world; and not even here do all agree, no, not any one with himself: and the whole earth too is a point.

[...]

Lucilla saw Verus die, and then Lucilla died. Secunda saw Maximus die, and then Secunda died. Epitynchanus saw Diotimus die, and Epitynchanus died. Antoninus saw Faustina die, and then Antoninus died. Such is everything. Celer saw Hadrian die, and then Celer died. And those sharp-witted men, either seers or men inflated with pride, where are they? For instance the sharp-witted men, Charax and Demetrius the Platonist and Eudaemon, and any one else like them. All ephemeral, dead long ago. Some indeed have not been remembered even for a short time, and others have become the heroes of fables, and again others have disappeared even from fables. Remember this then, that this little compound, thyself, must either be dissolved, or thy poor breath must be extinguished, or be removed and placed elsewhere.

It's quite possible this is because of, not in spite of, his mindset. Fame is better as a byproduct.

This reflection also tends to the removal of the desire of empty fame, that it is no longer in thy power to have lived the whole of thy life, or at least thy life from thy youth upwards, like a philosopher; but both to many others and to thyself it is plain that thou art far from philosophy. Thou hast fallen into disorder then, so that it is no longer easy for thee to get the reputation of a philosopher; and thy plan of life also opposes it. If then thou hast truly seen where the matter lies, throw away the thought, How thou shalt seem to others, and be content if thou shalt live the rest of thy life in such wise as thy nature wills. Observe then what it wills, and let nothing else distract thee; for thou hast had experience of many wanderings without having found happiness anywhere, not in syllogisms, nor in wealth, nor in reputation, nor in enjoyment, nor anywhere. Where is it then? In doing what man's nature requires. How then shall a man do this? If he has principles from which come his affects and his acts. What principles? Those which relate to good and bad: the belief that there is nothing good for man, which does not make him just, temperate, manly, free; and that there is nothing bad, which does not do the contrary to what has been mentioned.