In addition to local addressing (see MAC address), L2 is doing the framing - so essentially, no, you can't get away without L2.
You can avoid addressing by using point-to-point links only - one interface per peer, and likely a lot of meshing (forwarding nodes), but there's no alternative to framing.
Of course, you can use different names in your network model but you can't do away vital functions.
Can you consider a LAN to exist without layer 2 and just layer 1?
You could put a switch in the middle of your bench and call it a "LAN". That's just physical layer and data link layer, but as soon as you connect hosts and begin to make use of that "LAN" you add higher layers, at least an application layer.
If you place a repeater hub or a coax cable on your desk as "LAN", that'd be just the physical layer, no data link layer. Again, when you connect hosts you add the same layers as above.
Effectively, it all boils down to your definition of "LAN". In my book, a hypothetical LAN requires at the very least the physical, data link, and application layers. The transport layer may be present but it might be barely noticeable. Practically, you'd also a network layer for commonly expected functionality - then you've got the full stack.