Hm, our standard features can only support up to three menus (using "Variants affecting price/weight/stock/sku") plus three more fields which can be either menus or text-entry fields (using "Variations which tell more..."), making a total of 6 variables per product.
You can also add one more "custom" field/menu. Although that variable would be passed into the cart with any particular product's Add to Cart button, the custom value actually gets attached to the full order rather than any specific product in the order, and as such it can only be set once for the full order.
With some additional custom scripting, you could have additional menus/fields on the page that would get concatenated (strung together) into a single variable field to be passed into the cart.
This and the Custom variable are explained more in the Changing Behavior section here:
You may wish to consult with a competent, E-junkie-experienced developer regarding the more technically-involved, custom approaches described above: