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Saint Francis of Assisi did this. He renounced his family name, inheritance, and (according to legend) the clothes on his back when his dad took issue with him giving alms. He spent the rest of his life wandering with a small group of other penitents and providing comfort to the sick, especially those with leprosy.
Saint Benedict Joseph Labre followed a similar path. Though he was from a wealthy family, he strove to live a monastic life. When he was turned down twice, he resorted to becoming a homeless pilgrim who traveled between European holy sites until he died of starvation. Notably, though, he was said to avoid people who were too fond of him and practically sought out opportunities to be downtrodden.