One option is to obtain frequencies of all the combinations of product purchases; select the few most common combinations; then build a regression model to predict each individual's chosen combination. E.g., with a binary logistic regression you could conceivably predict purchase of a) White Wine, Brie, Strawberries and Grapes vs. b) Red Wine, Cheddar and Gouda. With more than 2 such combinations, or if you want to include the category of "none of the above," multinomial logistic regression would probably be the method of choice.
Note that including just the common combos means you will have more workable numbers of each but that you will be excluding the others, at least from this procedure. I could imagine 7 items creating dozens of combos each chosen by at least a few people. This is possibly too many categories for your sample size. Moreover, if a combo were chosen by just a few people, your model would have very little information to work with.
Another option is to use cluster analysis to arrive at a few sets of items that tend to be purchased together. With 7 items, you'll probably end up with fewer than 4 clusters, which might make your task easier. If you try cluster analysis and find the results unworkable, there is no reason why you have to use them: just go back to the frequency-based approach described above. In this case, if I read you right, you're looking for the most descriptive and interesting array of categories, and in establishing that, you don't need to worry about degrees of freedom or multiple comparisons or any such concerns that might apply if you were trying out multiple methods in performing some inferential test.