If you are looking for a hard and fast answer, there is none. Life does not have a hard and fast definition, so it is impossible to identify something that everyone will recognize as both living and non-cellular. However, your question is answerable if you are interested in a summary of the state of the debate.
- All living organisms are made up of one or more cells
An exception to this would be known as non-cellular life. Whether anything could or does fill this category is a controversial subject. Considering the fact that the main points included in most definitions of life are based on observations of cellular life, this is not surprising.
That said, the strongest candidates for non-cellular live (in descending order) include: viruses, viroids, and prions.
- Viruses are the non-cellular entities most often considered potentially alive. Viruses are even classified in genera and species. They carry genetic material, they evolve, and they use various proteins as their machinery of action. However, they are unable to metabolize or produce their own proteins. They depend on infected cells doing all the work for them. Because they are incapable of acting on their own, they are not considered living by many. (The definition for life usually includes metabolism, the ability to process one's own food and energy.) The counter argument is that all life ultimately depends on external factors for survival, so viruses depending on cells for living may not be a disqualifier since they still control the processes resulting in their reproduction.
- Viroids, like viruses, depend on other cells to carry out metabolic and reproductive processes for them. Unlike viruses, viroids lack any protein whatsoever. As a result, even fewer people are willing to consider viroids alive. However, the same logic used to argue the biotic nature for viruses can be used for viroids. They copy themselves the same way many RNA viruses do, with the exception being they forgo directing the construction of any protein coat for themselves. Conversely, viroids can also be used as evidence against even viruses being considered life, precisely because of their similarity. If a viroid obviously is not life, viruses must be held suspect since they can be argued similar to viroids than cellular life.
- Prions are interesting. They do not reproduce like viruses or cell. That is to say they do not direct cellular machinery to construct copies of themselves. Instead, they modify existing proteins they come in contact with. From one angle, they are far less active in their flavor of reproduction than viruses. However, they are also more directly involved in their self-replication than viruses. Also, they can spread between organisms. Mad cow disease is a well known example of this.
Back and forth arguments aside, it has become fairly common to view viruses as a sort of middle ground between life and non-living processes, not one or the other.
- Cells are the smallest units of life
This one is not quite separate from numbers 1 and 3.
- Any non-cellular lifeform could be used to ague that the cell is not the most basic unit of life.
- Similarly, a cell not originating from a pre-existing cell could be used to make a similar argument, because it is highly unlikely that all the process found in even the most basic cell could start from scratch. They would have needed to exist before the cell in question, and then become incorporated into the cell. This rational is the basis for hypotheses like the RNA world, which are used to explain how life's processes could have been started, in piecemeal fashion, before being locked in cells.
Of course, we can also used the above points to argue that the cell is, indeed, the smallest unit of life.
- Viruses, viroids, and prions can be used to support the assertion that cells are the smallest unit of life. Even though they are not cells, their reproduction is only made possible by the activities carried out within cells.
- Things like the RNA world (if real) only would have required the processes for life gradually building up from less ordered, spontaneous reactions. An RNA world could be argued as a world devoid of true life, since life can be argued as things which carry out several key processes, and not as merely the individual processes themselves.
- Cells arise from pre-existing cells
This one is the only easy one. This had to be violated by abiogenesis. There had to be a first cell (or group of cells created under common conditions).
The origins of the first cell(s) are variously explained by the RNA world, GADV-protein world, and other hypotheses. Whatever the case, however life started, it had to have a start. This means there must be one or more cells in the history of life that were spawned from some natural processes lacking a parent cell.
While it may be safe to say that all cells in history are not required to have had parent cells, it is also safe to say that it is extremely uncommon for parentless cells to come into being. Firstly, it is not something we have ever observed. Secondly, all indications point to all species being part of the same "tree of life." This would mean that whatever conditions spawned the first cells must not have persisted. Either all life descend from the same single cell, or life started with many independently spawned cells that blurred their independent origins due to heavy horizontal gene transfer.
We can view the cells from cells rule a bit like Newtonian gravity. It is mostly right, with its faults only becoming ovbvious under conditions not readily present on Earth.
Aseptate fungal hyphae, giant algae and skeletal muscle fibres are often used as examples of cells that do not conform to the cell theory
I am aware that the listed cells are unlike the "standard cell" ...
I forgot to answer this part, and I think it is important to do so. All of those conform to cell theory. Whether they conform to the pattern common to most cells or not is irrelevant. They all have the components required to classify them as cells. Cells merging into multi-nucleated cells (like skeletal muscle fibers) is actually a good argument for cells being the basic unit of life. Even though they start separate, they collaborate by becoming one cell. Sure, the resultant cell has multiple nuclei, but most cells on Earth have no nuclei.