There is one grid polygon of area $1$ ($1\times 1$ square), one such polygon of area $2$ ($1\times 2$ rectangle), $2$ such polygons of area $3$ ($1\times 3$ rectangle and an angle of $3$ squares). To satisfy the condition one must then use at least $4$ grid polygons of area $4$ or greater. Hence the area of the given rectangle $R$ is at least $1+2+2\cdot3+4\cdot4=25$. Observe that it cannot be exactly $25$. Otherwise $R$ is a $5\times 5$ square or a $1\times 25$ rectangle. The first case is excluded by hypothesis. In the second only rectangles $1\times k$ can be used in the division, hence $R$ would have area at least $1+2+...+8>25$. In conclusion, $R$ has at least area $26$. It can be exactly $26$ for a $2\times 13$ rectangle.

| 1 | 3 | 3 | 4 | 5 | 5 | 5 | 6 | 6 | 7 | 7 | 7 | 7 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 |

The figure displays a division of such a rectangle into $8$ different grid polygons. So the required minimal area is $26$.