He forgets things. Of course he forgets. He's much too old to hold onto every moment of his existence, and sadly, that means that many of his memories are blurred, distorted. But he doesn't forget their names. Nor does he forget their significance to not only him, but to the universe. They are his pride. Ordinary people, not turned into weapons as someone once said to him, but into the giants he always knew them capable of being.
But yes, he forgets things. He forgets their favorite colors, and the way they take their tea. He forgets many little things about them but he doesn't forget the overwhelming love he has for each of them. He loves them all, each in their individual ways.
There are certain things he desperately clings to, memories that are too important to forget. He remembers the holler of Donna Noble saying, "Oi! Spaceman!" He recalls Jack's overly flirty attitude with anyone or anything that breathes, and some things that don't. Sometimes, he can picture perfectly how Martha refused to call him the Doctor at first, telling him he had to earn that title.
Most of all, he remembers Rose. He remembers her tongue in tooth smile that would leave him beaming, and her thundering loyalty. When the universe said her time with him was over, she refused to stand down. In all rights, she truly was the Bad Wolf, truly a force to be reckoned with.
Though as much as she'd wanted to stay with him, there was another life waiting for her, a life she could start with the meta-crisis Doctor. And so she was merely nothing more than a memory to him as well. A fond memory, but a memory nonetheless.
Being a time traveler, the lives of everyone he's ever known, and ever will know, are relative. One moment they are 12 years old and then, simply with a trip in the TARDIS, they're all grown. They're living a slow life but he could watch their lives in an instant, from beginning, to end.
Somewhere, in one moment, Donna is happily married and without any memory of him. In another, she and his tenth self are solving a mystery during the 1920s. And in another, she's been dead for years and years and years. He could visit her grave. He could. But he won't.
The same applies to practically everyone he's ever known.
But it doesn't apply to Rose.
He'll never find her headstone in a cemetery. At least, it won't be real. Her ashes will not fly through the breeze of one of the planets they'd visited, nor will there be an accurate funeral that he can attend. Her death in this universe is nothing more than conspiracy.
And sometimes, when he's saddened by all those he will not visit, he's gladdened that there is no way for him to visit Rose's grave. Her body is not buried anywhere in the universe. It is not a pile of ash. He thanks the universe for once that Rose is not in it, for he could not bear to know her death. He can live in one fantasy where Rose is forever alive, because he will never see her die.
Yes, he forgets things. He forgets the little quirks that once made him smile. He can't recall the way they danced. But he doesn't stop loving them with both his hearts. He is a sentimental old man who has lived much too long, seen far too much, but loved more than should be possible. Even when his memory blurs, he recalls that they were all extraordinary in their own right.