You can use up all 150 slots, with exactly 8 people per team, as follows:

    Day 1:
    Team 1 {1,2,3,4,5,6,7,9}
    Team 2 {1,2,3,4,5,6,7,9}
    Team 3 {1,2,3,4,5,6,7,8}
    Team 4 {1,2,4,6,7,10}
    Team 5 {1,2,3,4,5,6,7,8}
    Team 6 {1,2,3,4,5,6,7,9}
    Team 7 {1,2,3,4,5,6,7,10}
    Team 8 {1,2,3,4,6,9,10}
    Team 9 {1,2,3,4,5,6,7,10}
    Team 10 {1,2,3,4,5,6,7,10}
    Team 11 {1,2,3,4,5,6,7}
    Team 12 {1,2,3,4,5,6,7,8}
    Team 13 {2,3,4,5,6,7,9}
    Team 14 {1,2,3,4,6,7,9}
    Team 15 {1,2,3,4,5,7}
    Team 16 {1,2,4,5,6,7,8}
    Team 17 {1,2,3,5,6,7,9}
    Team 18 {1,2,3,4,5,6,7,10}
    Team 19 {1,2,3,4,5,6,7,8}
    Team 20 {1,2,3,4,5,6,7,10}

    Day 2:
    Team 1 {2,3,4,5,6,7,9}
    Team 2 {1,2,3,5,7,9}
    Team 3 {1,2,3,4,5,6,7,8}
    Team 4 {1,2,3,4,5,6,7,10}
    Team 5 {1,2,3,4,5,7,8}
    Team 6 {1,2,3,4,6,7,9}
    Team 7 {1,2,3,5,7,10}
    Team 8 {1,2,3,4,5,6,9,10}
    Team 9 {1,2,3,4,5,6,7,10}
    Team 10 {1,2,3,4,5,6,7,10}
    Team 11 {1,2,3,4,5,7,9}
    Team 12 {1,2,3,4,5,6,7,8}
    Team 13 {1,2,3,4,5,7,9}
    Team 14 {1,2,3,4,5,6,7,9}
    Team 15 {1,2,3,4,5,7,8}
    Team 16 {1,2,3,4,5,6,7,8}
    Team 17 {1,2,3,4,5,6,7,9}
    Team 18 {1,2,3,4,5,6,7,10}
    Team 19 {1,2,3,4,5,6,7,8}
    Team 20 {1,2,3,4,5,6,7,10}

    Day 3:
    Team 1 {1,2,3,4,5,7,9}
    Team 2 {1,2,3,4,5,6,7,9}
    Team 3 {1,2,3,4,5,6,7,8}
    Team 4 {1,2,3,4,5,6,7,10}
    Team 5 {1,2,3,4,5,6,7,8}
    Team 6 {1,2,3,5,6,7,9}
    Team 7 {1,3,4,5,6,7,10}
    Team 8 {2,3,4,5,6,9,10}
    Team 9 {1,2,3,4,5,7,10}
    Team 10 {1,2,3,4,5,6,7,10}
    Team 11 {1,2,3,4,5,7,9}
    Team 12 {1,2,3,4,5,6,7,8}
    Team 13 {1,2,3,4,5,6,7,9}
    Team 14 {1,2,3,4,5,6,7,9}
    Team 15 {1,3,4,5,6,7,8}
    Team 16 {1,3,4,5,6,7,8}
    Team 17 {2,3,4,5,6,7,9}
    Team 18 {1,2,3,4,5,6,7,10}
    Team 19 {2,3,4,5,6,7,8}
    Team 20 {1,2,3,4,5,6,7,10}

    Day 4:
    Team 1 {1,2,3,4,5,6,7,9}
    Team 2 {1,2,3,4,5,6,7,9}
    Team 3 {1,2,3,4,5,6,7,8}
    Team 4 {1,2,3,4,5,6,7,10}
    Team 5 {1,2,3,4,5,6,7,8}
    Team 6 {1,2,3,4,5,6,7,9}
    Team 7 {1,3,4,5,6,7,10}
    Team 8 {1,2,5,6,9,10}
    Team 9 {3,4,5,6,7}
    Team 10 {1,2,3,4,5,6,7,10}
    Team 11 {1,2,4,5,6,7,9}
    Team 12 {1,2,3,4,5,6,7,8}
    Team 13 {1,2,3,4,5,7,9}
    Team 14 {1,2,4,5,6,7,9}
    Team 15 {1,3,4,5,6,7,8}
    Team 16 {1,2,3,4,5,6,7,8}
    Team 17 {1,2,3,4,5,6,7,9}
    Team 18 {1,2,3,4,5,6,7,10}
    Team 19 {1,2,3,4,5,6,7,8}
    Team 20 {1,2,3,4,5,6,7,10}

    Day 5:
    Team 1 {1,3,4,5,7,9}
    Team 2 {1,2,3,4,5,6,7,9}
    Team 3 {1,2,3,4,5,6,7,8}
    Team 4 {1,2,3,4,5,6,7,10}
    Team 5 {1,2,4,5,6,7,8}
    Team 6 {2,3,4,5,6,7,9}
    Team 7 {1,2,3,4,5,6,7,10}
    Team 8 {1,2,3,4,5,6,9}
    Team 9 {1,2,3,4,5,6,7,10}
    Team 10 {2,3,4,5,6,7,10}
    Team 11 {1,2,3,4,5,6,7,9}
    Team 12 {1,2,4,5,6,7,8}
    Team 13 {1,2,3,4,5,6,7,9}
    Team 14 {1,2,3,4,5,7,9}
    Team 15 {1,2,4,5,6,7,8}
    Team 16 {1,2,3,4,5,6,7,8}
    Team 17 {1,2,3,4,5,6,7,9}
    Team 18 {1,2,3,4,5,6,7,10}
    Team 19 {1,2,3,4,5,6,7}
    Team 20 {1,2,3,4,5,6,7,10}

I obtained the solution via integer linear programming, and 8 is the minimum possible value of the largest team.