I'm not sure how practical this is for your given requirements, but you can try bitwise algebra. Given that you have a limited number of species, it might work.
Bitwise algebra would work where the species are all represented by either 1, 2, 4, 8, 16, or a combination of those. For example, is species 1, 3 and 4 exist the cell would be represented by
1 + 4 + 8 = 13. You can then use further bitwise operations to reverse that query. Example
13 AND 1 = 1,
13 AND 4 = 4,
13 AND 2 = 0. Where the evaluation of a bitwise AND is
> 0, the test for comparison is true.
You can read more on ESRI's implementation of bitwise here. The ESRI Raster Calculator has bitwise operators. Most other GIS tools should have them as well.
However, given that you will eventually encounter a scenario where you will have more numerous species you might want to think of another method. Ideally, from an ecological perspective, you would need to identify specific groups of communities, or biodiversity categories to represent the combinations of data. (See ChrisW's answer on using the Combine tool)
Alternatively, you would want to keep your layers separate and build a system around it that queries each layer independently if you can't establish categories.