Llywelyn ap Iorwerth, born around 1172, was the grandson of Owain Gwynedd and ruled Wales from the late 12th century (certainly by 1200) to his death in 1240 AD.  He married Joanna (Joan), the eldest (albeit illegitimate) daughter of King John of England.

Llywelyn “proved to be the greatest and most constructive Welsh statesman of the Middle Ages. In his long career he succeeded, by constant warfare, by tactful yielding under pressure and by masterly resilience the moment that pressure was relieved, in bringing under his control most of Pura Wallia. When he died in 1240, full of honor and glory, he left a principality which had the possibility of expanding into a truly national state of Wales. There was a moment when an independent Wales seemed about to become a reality.”  http://www.castlewales.com/llewelyn.html

The Chronicle of the Princes (Ystrad Fflur edition) details the events of the 13th century in more detail than virtually any other contemporary source, particularly from a Welsh perspective, albeit one written by monks.   Llywelyn’s conquests are treated with some detail and give insight into the kind of ‘constant warfare’ to which the above quote refers:

1211 In this year Llywelyn ab Iorwerth led frequent attacks against the Saxons, harassing them cruelly. And because of that, John, king of England, gathered a mighty host and made for Gwynedd, planning to dispossess Llywelyn and to destroy him utterly. And the king came as far as Chester and to the castle of Degannwy. And there the host suffered lack of food to such an extent that an egg was sold for a penny-halfpenny; and they found the flesh of their horses as good as the best dishes. And because of that the king having lost many of his men, returned in shame to England without having fulfilled aught of his mission. And he returned again in August, and with him a host which was greater and fiercer.  And Llywelyn, being unable to suffer the king’s rage, sent his wife, the king’s daughter, to him by the counsel of his leading men to make peace with the king on whatever terms he could. And after he had accepted safe conduct to go to the king and to come away from him free, he went to the king and was reconciled to him. And then all the princes of Wales made peace with the king, except the two sons of Gruffudd, son of Yr Arglwydd Rhys. And the king with great joy and victory returned to England.  And he commanded Falkes, sheriff of Cardiff, to take all the host of Glamorgan and Dyfed with him to force the sons of Gruffudd ap Rhys to yield or else to drive them from all the kingdom. And Rhys and Owain, being unable to counter such great might as that, sent messengers to Falkes to draw up peace for them; for there was no place for them to flee in all of Wales. And Rhys and Owain went to the king under safe conduct of Falkes; and the king received them into reconciliation and into peace.

1212 In this year Llywelyn ab Iorwerth, prince of Gwynedd, being unable to bear the injuries which the men from the new castles were inflicting upon him, made a solm pact with the princes of Wales, namely, Gwenwynwyn, Maelgwn ap Rhys, Madog ap Gruffudd Maelor, Maredudd ap Rhobert. And he rose up against the king, and by the end of two months, he laid seige to all the castles which the king had built in Gwynedd, and took them all except two, Degannwy and Rhuddlan.  And three leaders of gentle birth from Wales were hanged in England, namely, Hywel ap Cadwallon, Madog ap Maelgwn, Meurig Barach.  And Pope Innocent the Third absolved three princes, namely, Llywelyn ab Iorwerth and Gwenwynwyn and Maelgwn ap Rhys, from the oath and allegiance they owed to the king of England. And he enjoined upon them, for the remission of their sins, to direct friendly endeavour and action against the iniquity of that king. And he interdicted the churches for five years in all England and Wales, except for the territory of those three princes and those who were leagued with them.

1213 In this year John, king of England, went to the archbisho of Canterbury to do penance. And he recalled the archbishop and the bishops and the clerics who had gone into exile because of the interdict on the churches. And he swore, too, that he would restore everything that he had taken from the Church.  And Llywelyn ab Iorwerth took the castle of Degannwy and the castle of Rhuddlan, and he gained possession of them.

The included map shows the lands Llywelyn Fawr controlled directly (yellow) and those belonging to his client princes (gray) circa 1271 AD.