Fuller won the Australian Junior Chess Championship in 1963, and represented Australia at the World Junior Chess Championship in Vrnjačka Banja, Yugoslavia later that year. He finished equal first with Trevor Hay in the 1972 Australian Chess Championship in Melbourne. The play-off match between them was tied 5-5, so they were declared joint champions, a relatively rare occurrence. In the 1972 Asian Zonal Championship in Hong Kong, he finished equal second with Kan and Tan, behind Eugenio Torre. After the play-off match with Tan and Kan was also tied, the second qualifying place into the Interzonal was decided by a random drawing of lots. If Fuller had chosen the right envelope, he would have automatically gained the IM title and the Interzonal place, but Tan Lian Ann chose the correct envelope and went on to the Interzonal in Petrópolis, Brazil. In the 1974 Asian Team Chess Championship in Penang, Fuller played on board one for Australia and scored 6/7, winning the individual gold medal. Fuller won the Australian Open in 1973 and 1975, and the Doeberl Cup in 1974, 1975 and 1983. He won the New South Wales state championship in 1965, 1986, and in 1988. He won the NSW Lightning Chess Championship in 1967, 1970, 1971, 1972, 1973 and 1986, and the City of Sydney Championship in 1964, 1965, 1966, 1971 and jointly with Flatow and Levin in 1981. Fuller shared second place at the British Chess Championship in Coventry 1970, and in Morecambe 1975, as the events were also open to citizens of any Commonwealth country. Fuller represented Australia in nine Chess Olympiads between 1964 and 1990, usually on board 2, scoring 59 points from 106 games. His Olympiad victories include wins against IM Lazaros Viantiadis, IM Peter Biyiasas and IM Tüdeviin Üitümen, IM Juan Manuel Bellón López and IM Leif Øgaard. He was the captain of the Australian team at the 1986 and 1990 Chess Olympiads. In the 1970s, Maxwell Fuller lived for some years in London. In the late 1990s he discontinued chess activity, partly due to health problems. Fuller returned to chess in 2004 after an eight-year absence and finished equal second in the 2004 and 2005 NSW championships, before finally retiring from FIDE rated chess tournaments. His highest FIDE rating was 2400 in 1975.
