*theoretically

Part 1
Lions needed: 2

• Lion A (male, make king)


• Lion B (female)

1. Breed Lions A and B to make lion C.


2. Replace your king, Lion A, with Lion C

Part 2
Lions needed: 2

• Lion C (male, king)


• Lion B (female, from previous step)

1. Breed Lions C and B to make Lion D (female).

Part 3
Lions needed: 2

• Lion C (male, king)


• Lion D (female, from previous step)

1. Breed Lion C with Lion D to make 2 cubs, E (male) and F (female)


2. Replace your king, Lion C, with Lion E.

Part 4
Lions needed: 2

• Lion E (male, King)
  
• Lion F (female)

1. Breed Lion E and Lion F to make Lion G (male) and Lion H (female)
   
2. Replace your King, Lion E, with Lion G

Part 5
Lions needed: 2

• Lion G (male, King)


• Lion H (female)

1. Breed Lions G and H to make Lion I, their Super Inbred Baby.

I think my math is right, at least. theoretically this should make the most inbred lion possible, though i have no idea how many instances of inbreeding it would have.