Is it the the Great Commandments, or perhaps the New Commandment in John?
No, let me suggest that the most radical ethical claim in the Bible is, in fact, Mark 8:34 (and its parallels in Luke 9:23 and Matthew 16:24; the wording varies slightly, but in no significant details).
If anyone would come after me, let him deny himself and take up his cross and follow me.
Jesus' startling statement, primarily to his disciples, but with an eye to anyone who would be a disciple, comes in all three Synoptics after the Caesarea confession of Peter, a Messianic insight granted by the Father, which represents a turning point in each of the gospels, most distinctively in Mark. Jesus, immediately following Peter's confession, initiates a new stage in his teaching, as in Mark 8:31-32. For Jesus, his messianic mission is inextricably linked to suffering, death, and resurrection. And it is this that he makes the decisive pattern for discipleship.
It is proverbial to understand 'take up one's cross' as 'accept one's lot of suffering in life', as if the cross here were metaphorical. Nothing could be further from the truth. In Jesus' context, the cross means but one thing: public execution by the ruling authorities, as a non-citizen of no account. That meaning is only confirmed by the gospel story itself.
The radical claim then, is this: to be a follower of Jesus will involve, firstly the denial of self. This is not 'to deny X to Y', as in 'to deny icecream to myself'. The Christ-follower does not live a life characterised by denial-of-things-to-self, though that too may become a part of it. No, here is a far more thorough, radical, denial-of-self. It is the denial of knowledge, of allegiance, of determination, and of self-determination. It is to say, "I deny myself", much as Peter will later deny Jesus. It is to say, "I no longer determine my self-identity", the ultimate refutation of an existentialist humanism that sees the only real meaning as self-determined meaning. It is to ultimately and utterly surrender the self to the determination of another, to the determination of the one followed, the crucified Lord.
The radical claim is followed by a radical paradigm: take up one's cross. Jesus-followers commit to a way of life that led Jesus to the cross, to rejection by the world, to social ostracisim, to political martyrdom. However often you are told that Jesus' messianic kingship was "spiritual, not political", we must be reminded that John 18:36 teaches us that Jesus' kingdom, while not of this world, is very definitely in this world, even as he was and his disciples are, as he himself prays in John 17:11. The allegiants of Jesus' kingdom, as he did, will find themselves in conflict by their very nature and allegiance with the kings of this world, the Herods, Pilates, Caesars, Ceausescus, Stalins, Hitlers. 1 Peter 2:21 lays down the same message, the same pattern, that our ethical, social, political existence is patterned on the Suffering Servant of Isaiah 53, Jesus of Nazareth.
Only the radical commitment to deny ourselves, to take up our crosses, to follow Jesus, gives us the new identity, the Jesus-determined identity, to trust our heavenly father, await his just judgment (1 Peter 2:23), love our enemies as they pierce our side, and witness to the truth (John 18:37).
That's why Mark 8:34 is the most radical ethical claim in the Bible