The maximum amount of memory that can be installed in an LC III / LC III+ / Whatever else Apple stuffed that logic board in is 36 MB. 4 MB onboard and a single 32 MB SIMM. The logic board does not support 64 or 128 MB SIMMs due to lack of address lines. If such a module is installed, the machine either won't boot or will show 32 MB or less, depending on the topology of the module.
36 MB of RAM was more than enough memory for this logic board, if you need more than that, you're probably exceeding the processing capabilities of the CPU. You can use RAM Doubler 9 to increase the amount of memory via compression and virtual memory, but it will considerably slow the machine down.