While I appreciate you thinking in a different way, I would say that is not a good idea to have a linked 'list' without 'list'. A linked list is basically a list of Nodes linked together. When there is no 'List' class where do you plan to keep the reference to the Head and/or End of the linked list? And where do you plan to implement the Linked List specific functionalities such as traversal, insertion, deletion etc.? Without a 'List' eventually you would end up in implementing a procedural oriented linked list program in C++.
The link that 'OriginalGriff' has shared shows the implementation of a linked list in "pure C". While it is a good start to understand the basics of linked list, it is not the best way to implement a linked list in C++.
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the Node class should have *prev and *next and should manage the allocation and de-allocation of the *prev and *next objects.
Of course the prev and next pointers should be a part of the Node class. But its not the duty of the Node class to allocate memory for them. Actually *prev and *next are pointers to link a Node to its previous and next Nodes respectively. If a Node itself is responsible for allocating and de-allocating its prev and next pointers, then it would be like a Node creating its previous and next Nodes. In that case the most basic question would be 'Who is responsible for creating the first Node?'