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Bigger was born in Little Donegal Street in Belfast in 1863. He was the seventh son of a seventh son. His ancestors came from the lowlands of Scotland in the 1630s, part of a consortium of merchants who bought their own trading ship. They named it the Good Ship Unicorn of Belfast and it was one of the first trading ships ever owned by local merchants. Bigger was educated at the Royal Belfast Academical Institution, known as "Inst". His grandfather was a founder of the school in 1810 and his father was a governor.

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The breadth of his interests was astonishing. He joined the Belfast Naturalists' Field Club, a diligent body of men founded in 1863 that boasted amongst its membership the photographer R.J. Welch, the botanist Robert Lloyd Praeger and Ralph Tate, the author of Flora Belfastiensis, the first study of Belfast flora. Bigger became secretary and later president of the BNFC. Apart from botany, archaeology was one of his passions, and he revived the Ulster Journal of Archaeology after an absence of 30 years, becoming its editor. In 1911 his interest in archaeology led him to buy the 15th-century Jordan's Castle at Ardglass in south Down. He restored the castle using it as a weekend retreat and later bequeathed it to the state.

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Although his archaeological writing was well regarded by the scholars of his day, subsequent generations of archaeologists were less enthusiastic. Some felt his approach was more that of a romantic historian rather than a serious archaeologist. His purple prose from a survey of the Friary of Bun-na-Margie near Ballycastle in north Antrim illustrates the point:

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"We can readily picture the friars, on a bright Easter morning, trooping out of the beautiful church after the early service, with the resounding Te Deum and the music of the songs of exultation still ringing in their ears, to gaze up at the great dome of Knoc-lade clearly cutting into the blue sky, the fleecy clouds chasing each other like lambs across the valley of Glenshesk, with the winding waters of the Margie dancing over their pebbly bed in the sunlight close at hand. . ."

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His connection with the Glens of Antrim was particularly strong. He became involved in organising the Glens Feis, which grew out of his desire to hold a series of large Gaelic cultural events. This turned into a pageant, with pipers in Elizabethan costume and a celebration of all aspects of Irish culture from language to basket-weaving. Bigger also developed a lifelong interest in the Irish language and the Gaelic League, later becoming a member of the league's executive committee.

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Through his involvement with it he met the league's president, Douglas Hyde as well as Roger Casement, and both became friends of his. They were frequent visitors to his home, Ardrigh House, on Belfast's Antrim Road. Another visitor was the musician Francis McPeake, who wanted to play the uileann pipes and revive what was then a dying art. His association with Bigger became a legend in Irish traditional music circles. Bigger arranged for John O'Reilly, a blind piper from Galway, to come and stay in Belfast and teach McPeake to play. O'Reilly stayed for three months with the McPeakes. Bigger gave him five shillings a week spending money, 15 shillings a week to send home to his family, and seven shillings and sixpence to the McPeakes for his lodging.

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Another of his enterprises was improving the standards of pubs. He founded the Ulster Public House Association with the purpose of cleaning up disreputable pubs and turning them into convivial inns with clean and safe surroundings. One such establishment was the Dunleath Arms in Ballywalter on the Co Down coast which is now a B&B and survives in the Arts and Crafts style in which it was built.