10 Consecutive NBA Appearances - Bill Russell
Bill Russell holds the record for the most consecutive NBA final appearances, at 10! From first joining the Boston Celtics in 1956, Bill made it to the NBA finals from 1956 to 1966. After winning his first NBA championship in 1957, Bill Russell went on to win the NBA championship 8 years in a row with the Celtics, from 1959 to 1966. In 1968 and 1969, we proceeded to win two more championships as a player-coach.
9 Consecutive NBA Appearances - Sam Jones & Tom Heinsohn
Sam Jones played for the Boston Celtics from 1957 to 1969, going on the same 8-year winning streak as the aforementioned Bill Russell, and ending his season with 10 NBA championships, two more from 1968 and 1969.
Tom Heinsohn, another Boston Celtic and teammate of Bill Russell an Sam Jones also appeared 9 times in a row at the NBA finals, winning the championship 7 consecutive years from 1959 to 1965. He went on to win two more NBA championships as their coach.
8 Consecutive NBA Appearances - K.C Jones, Frank Ramsey & Lebron James
K.C Jones played for the Boston Celtics from 1958-1967, going on to appear at eight consecutive NBA finals from 1958-1966 and winning every NBA final from his rookie season. Additionally he won two NBA Championships as a coach (1984 & 1986) and two championships as an assistant coach (1972 & 1981).
Frank Ramsey also played from the Boston Celtics from 1954-1955, then from 1956-1964. In that time he made 8 consecutive NBA final appearances, from 1959-1964, accumulating 7 total NBA championships.
LeBron James, as the OP mentioned, just made his 8th consecutive NBA finals, four with the Miami Heat from 2011 to 2014 and 4 with the Cleveland Cavaliers from 2015 to 2018, however he has only won 3 championships.