We find a new Tribonacci-like sequence of positive integers ⟨ x₀, x₁, x₂, ... ⟩ given by x_n = x_n-1 + x_n-2 + x_n-3, n ≥ 3, and gcd(x₀,x₁,x₂) = 1 that contains no prime numbers. We show that the sequence with initial values x₀ = 151646890045, x₁ = 836564809606, x₂ = 942785024683 is the current record in terms of the number of digits.