Summary: Paul gives three reasons for boasting only in the cross of Jesus Christ: 1) The cross has the power to free believers from the world’s bondage (Galatians 6:14b), 2) It has the power to do what the flesh cannot do, and 3) It has the power to bring salvation.
Galatians 6:14-18  But far be it from me to boast except in the cross of our Lord Jesus Christ, by which the world has been crucified to me, and I to the world.  For neither circumcision counts for anything, nor uncircumcision, but a new creation.  And as for all who walk by this rule, peace and mercy be upon them, and upon the Israel of God.  From now on let no one cause me trouble, for I bear on my body the marks of Jesus.  The grace of our Lord Jesus Christ be with your spirit, brothers. Amen. (ESV)
An almost ubiquitous feature of modern western culture is the proclamation of self-esteem. Kids are educated to think they can do whatever they set their minds to. People are told to take pride in their work, family, and country. This thought is not a new one. Greek philosophy reflecting Hellenistic culture, greatly valued all forms of human assertiveness as badges of excellence, strength, and virtue (from the Latin virtus, meaning “manliness” or “worth”). Physical prowess, military feats, oratorical abilities, intellectual acumen, political power, monetary success, social status—all these were things to be proud of and to glory in (George, T. (1994). Galatians (Vol. 30, p. 436). Nashville: Broadman & Holman Publishers.)
The Judaizers of Galatia likewise put all their weight in their works. Their works, like all human effort, cannot bear the weight of divine perfection required for standing before God. Their boast in the flesh (Gal. 6:13) is such a dangerous practice that Paul exposes the folly in the beginning of Galatians 6:14a: “But far be it from me to boast except in the cross of our Lord Jesus Christ, (by which the world has been crucified to me, and I to the world”). But far be it from me /May it never be (me genoito), translates a strong negative that carries the idea of virtual impossibility. He uses it here to tell the Galatians that it was inconceivable for him even to think of boasting in anything but “the cross of our Lord Jesus Christ”. Paul could have boasted in his learning, his literary gifts, his power as a preacher, his travels, his evident success as a missionary, the esteem in which he was held by countless believers, his high standing as an apostle, as well as many other things. But Paul sought and found joy and satisfaction in Christ (Edgar H. Andrews: Free in Christ. Welwyn Commentary Series. Evangelical Press. 1996. p. 326).
Paul, however, chose something utterly despicable, contemptible, and valueless to the modern culture as the basis of his own boasting—the cross of our Lord Jesus Christ. For two thousand years the cross has been so variously and beautifully represented in Christian iconography and symbolism that it is almost impossible for us to appreciate the sense of horror and shock that must have greeted the apostolic proclamation of a crucified Redeemer. Putting it into another cultural context, Clarence Jordan paraphrases this verse: “God forbid that I should ever take pride in anything, except in the lynching of our Lord Jesus Christ”. The Latin word for cross (crux) was regarded as an expression so crude no polite Roman would utter it in public. In order to get around this difficulty, the Romans devised a euphemistic circumlocution, “Hang him on the unlucky tree” (arbori infelici suspendito) (Bruce, Galatians, 271, quoting Cicero, Pro Rabirio, 13.) The Greeks also found the cross disgusting. While they revered the human body as a thing of utmost beauty, the cross mangled and shamed it. The Jews, as noted in Gal 3:13, considered the cross a curse. (Boles, Kenneth L.: Galatians & Ephesians. Joplin, Mo. : College Press, 1993 (The College Press NIV Commentary), S. Ga 6:14)
But what the world regards as too shameful to whisper in polite company, a detestable object used for the brutal execution of the dregs of society, Paul declared to be the proper basis for exultation. In this and in this alone he would make his boast, in life and death, for all time and eternity. (George, Timothy: Galatians. electronic ed. Nashville : Broadman & Holman Publishers, 2001, c1994 (Logos Library System; The New American Commentary 30), S. 435). Although kauchaomai (to boast) often refers to evil boasting (see, e.g., 1 Cor. 1:29; 3:21; Eph. 2:9), it here carries the idea of good glorying or rejoicing, as it does in Romans 5:2, 3, 11 (“exult”) and Philippians 3:3 (“glory”). By definition, the English word boasting is an expression of pride, which was the furthest thing from Paul’s intent. The Greek term carries the basic meaning of praise, and whether it represents a sin or a virtue depends on whether self or God is being praised. John Stott notes that it means “to glory in, trust in, rejoice in, revel in and live for. The object of our boast or glory fills our horizons, engrosses our attention, and absorbs our time and energy. In a word our glory is our obsession (John R.W. Stott, The Cross of Christ (Dowers Grove, IL; InterVarsity, 1986. p. 349).