A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.
To find the volume of the largest box that can be completely enclosed inside the cylindrical container with a hemispherical lid, we need to consider the dimensions of the container and the constraints they impose on the box.

1. **Determine the dimensions of the cylindrical container:**
   - The total height of the container (including the hemispherical lid) is \(60\) cm.
   - The radius of the base of the container is \(30\) cm.

2. **Calculate the height of the cylindrical part of the container:**
   - The height of the hemispherical lid is equal to the radius of the base, which is \(30\) cm.
   - Therefore, the height of the cylindrical part of the container is:
     \[
     \text{Height of cylindrical part} = 60 \, \text{cm} - 30 \, \text{cm} = 30 \, \text{cm}
     \]

3. **Determine the maximum dimensions of the box:**
   - The box must fit within the cylindrical part of the container, so its height must be less than or equal to \(30\) cm.
   - The base of the box must fit within the circular base of the container, so the side length of the square base of the box must be less than or equal to the diameter of the base of the container, which is \(2 \times 30 \, \text{cm} = 60 \, \text{cm}\).

4. **Calculate the volume of the largest box:**
   - The largest box that can fit inside the container will have a square base with side length equal to the diameter of the base of the container and a height equal to the height of the cylindrical part of the container.
   - Therefore, the side length of the base of the box is \(60\) cm and the height of the box is \(30\) cm.
   - The volume of the box is given by:
     \[
     \text{Volume} = \text{side length}^2 \times \text{height} = 60^2 \times 30 = 3600 \times 30 = 108000 \, \text{cm}^3
     \]

The final answer is \(\boxed{108000 \, \text{cm}^3}\).