At the opening of Rembrandt: The Late Works at the National Gallery (until 18 January), I met a painter friend of mine in the final room. This was, he said, one of the most magnificent exhibitions he had seen in his entire life, which — considering he is perhaps 70 and a frequent visitor of galleries — was praise indeed (and entirely deserved).

Mischievously, I mentioned that he had also been highly enthusiastic about Veronese at the National Gallery a few months ago. ‘Ah, but there is a huge difference between Veronese and Rembrandt,’ he vehemently responded. ‘When you look at a Madonna by Veronese, you see a glamorous model wearing expensive clothes, with Rembrandt’s “Bathsheba” [pointing at the picture in front of us] you can read her thoughts.’

He got it in one. Rembrandt is the supreme painter of the inner life. He brings you close to the people in his pictures. That is one of the lessons the exhibition aims to teach. It and the accompanying book — more a collection of essays than a catalogue — contain sections such as ‘Intimacy’, ‘Inner Conflict’ and ‘Contemplation’. The fundamental premise is that in about 1650–52, Rembrandt (1606–69) changed course, becoming a more idiosyncratic and deeply original artist than he had been before. ‘Late Rembrandt’ therefore begins in what we would now think of as his early middle age when the artist was around 45.

To make their point, the curators have assembled, if not quite all the greatest of Rembrandt’s later works, a sensationally large number of them (one or two more will be added when the exhibition moves to the Rijksmuseum, Amsterdam, next year). Room after room contains a stunning assortment of paintings, drawings and prints, harmonising wonderfully with each other.

Picking highlights is hard. The series of self-portraits right at the start is a show-stopper, so too is the room of portraits halfway around, and the extraordinary conjunction right at the end of ‘Jacob Blessing the Sons of Joseph’ (1656), a meditation on youth, age and family love, with ‘Bathsheba with King David’s Letter’ (1654).

The latter is as convincing an evocation of human flesh and skin as anything by Titian, with the difference that it seems like the real body of an individual woman, sitting with a slight and touching awkwardness. But that’s only part of what the picture does. She is gazing into space, anxious, perhaps fearful: not just a nude, but a person. The picture is not merely sensuous but also poignant and dramatic. My friend was right: you see what she is thinking.

As you walk through the exhibition you follow Rembrandt’s thoughts, too. Nowhere is that clearer than in the sequences of prints in different ‘states’ that hang side by side. This sounds a little arcane, but is actually gripping. Effectively, Rembrandt invented a new method of print-making.

By using drypoint — that is, pulling his needle through a copper plate so as to churn up a burr of metal on either side of the line — he was able to create luxuriously deep and velvety blacks. The disadvantage was that this burr quickly wore away so he could only make a limited number of impressions. Rembrandt’s reaction was to use this as an opportunity to re-cut, and radically rethink, the entire work, so ‘The Three Crosses’ (1653) is not a single image but a sequence of works — variations on a theme — culminating in a scene of apocalyptic darkness.

An 18th-century biographer, Arnold Houbraken, quotes Rembrandt as saying, ‘If I want to relieve my spirit, then I should seek not honour but freedom.’ Whether or not he actually said those words, they sound convincing. Freedom was what he sought in his painterly and graphic techniques, in the most audacious way.

His oil paintings were unconventional, in that he used extremely thick paint, loose and free brush-strokes, scratched with the handle of his brush, and slathered on pigment with his palette knife. But he didn’t have a set technical formula; he was nonconformist in a different way almost every time he painted a picture. ‘The Conspiracy of the Batavians under Claudius Civilis’ (about 1661-62) is rough-hewn to a startling extent, so that the faces of the Batavians plotting against their Roman masters by lamplight are conjured up by a few strokes of a loaded brush. This may have been why the monumental picture, commissioned to hang in Amsterdam’s Town Hall, was rejected.

In contrast, ‘The Wardens of the Amsterdam Drapers’ Guild’ (aka ‘The Syndics’, 1662), while its paint surface is also gnarled and crusted, is much more finely executed, perhaps because it was intended to be seen much closer up. Though a single stroke may have been put in at speed, the whole process was slow and considered. ‘The Syndics’ was not unusual in evolving through numerous arrangements until he felt he had got it right.

Thus Rembrandt’s works all ended up looking like Rembrandts, but each in an individual way. In a comparable spirit, Lucian Freud used to say his ambition as an artist was ‘not to have a method’. Perhaps Rembrandt’s example lay behind another of Lucian’s axioms: that fear of death is a great disadvantage for any painter.

Certainly Rembrandt didn’t seem to have one. On the contrary, he seems to fix on each new sign of his own physical decay — sagging jaw, puckering skin, bags under the eyes — as an opportunity to make another adventurous picture. The last ‘Self Portrait’ (1669) shows the flabby face of a man whose health is failing, but there’s no let up in the zest with which he records the pouches, wrinkles and bulbous drinker’s nose.

One of the most modern things about Rembrandt is that, again according to Houbraken, he insisted that the only criterion by which you could judge a painting finished was if the artist thought it was: ‘A work is finished when the master has achieved his intention in it.’ That could have been said by Picasso or Jackson Pollock.

It is impossible to decide whether ‘Self Portrait with Two Circles’ (1665–69), from Kenwood House (a strong candidate for the title of greatest painting in London, by the way), is finished or not. There are bits that are barely sketched — the hand holding the palette is barely there, though somehow you seem to see it. But you wouldn’t want to add a brush mark: more detail of the hand would distract from the face, and that’s what holds your attention. The look Rembrandt gives you from that 350-year-old canvas is truly formidable: there’s none more so in art.