You can rotate around the world axes in a custom order using Transform.RotateAround:

void Update()
{
   transform.rotation = Quaternion.Identity;
   transform.RotateAround(transform.position, Vector3.right, rotation.x);
   transform.RotateAround(transform.position, Vector3.up, rotation.y);
   transform.RotateAround(transform.position, Vector3.forward, rotation.z);
}

By nature, rotating an object will change its local axes. If you want a persistent set of "local axes", you can parent the gameobject to another gameobject, which holds the "local axes".

You can rotate around these axes in a custom order using Transform.RotateAround:

void Update()
{
   Transform par = transform.parent.transform;
   transform.rotation = Quaternion.Identity;
   transform.RotateAround(transform.position, par.right, rotation.x);
   transform.RotateAround(transform.position, par.up, rotation.y);
   transform.RotateAround(transform.position, par.forward, rotation.z);
}

You can rotate around the world axes in a custom order using Transform.RotateAround():

void Update()
{
   transform.rotation = Quaternion.Identity;
   transform.RotateAround(transform.position, Vector3.right, rotation.x);
   transform.RotateAround(transform.position, Vector3.up, rotation.y);
   transform.RotateAround(transform.position, Vector3.forward, rotation.z);
}

You can rotate around the world axes in a custom order using Transform.RotateAround:

void Update()
{
   transform.rotation = Quaternion.Identity;
   transform.RotateAround(transform.position, Vector3.right, rotation.x);
   transform.RotateAround(transform.position, Vector3.up, rotation.y);
   transform.RotateAround(transform.position, Vector3.forward, rotation.z);
}