Dominoes on a clipped board

Colour the cells like a chessboard and count each colour. Every domino covers one cell of each colour.

1. Colour the cells like a chessboard with the corners in the first colour. That colour has $25$ cells and the other has $24$, and the $4$ removed corners all have the first colour, leaving $21$ and $24$.
2. Each domino covers one cell of each colour, so at most $21$ dominoes fit.
3. A packing of $21$ exists: $3$ horizontal dominoes in each of rows $2$ to $6$ over columns $1$ to $6$ ($15$), $2$ in each of rows $1$ and $7$ over columns $2$ to $5$ ($4$), and $2$ vertical ones in column $7$ over rows $2$ to $5$.
4. The largest number is $15+4+2=21$.

Answer: $21$