When Elder Tales was first released, the maximum attainable level was 40. It wwas gradually increased throughout the expansion packs. Before the Catastrophe, the maximum attainable level was 90.
According to the Wiki,
With the release of the Homesteading the Noosphere expansion pack, the maximum level cap rose to 100. However, this change probably only applies to the Yamato server, as the other servers were not yet implemented with the expansion pack when the Catastrophe struck.
It is unclear to exactly how many servers was this expansion pack applied which implies that not all the players trapped inside the game have the ability to grow their levels beyond 90. But the expansion pack does apply to the Yamato server. Meaning the players in Log Horizon can definitely level up beyond 90. It can be concluded that leveling up beyond 90 is just very very slow and requires a lot of EXP points.
Moreover, it kind makes sense too that we don't see any characters above level 90 since none of them are actually seen grinding for high level monsters. Shiroe's gang was at doing that at least before Shiroe started working for the Round Table Conference. But they had no time for that after Shiroe tried to bring everyone together. They have become more "politics oriented". Trying to set up proper living system in Akihabara, the round table conference and all. None of them is really going all out for EXP. So it seems only logical that none of them were able to accumulate enough EXP points yet.
Here's a link to the level system resource from the Wiki.