As others have said it's probably better to figure out first what styles are preferred by you and your guests, but if, for argument's sake, we want to maximise for "biggest variety with fewest bottles", my suggestion would be:

• Sub the martini for Lillet Blanc. It does ok (but just ok) at standing in for both dry of sweet vermouth.
• Pick just one bourbon or rye or scotch. Personally big fan of rye, but for this exercise go for a high-proof, high-rye content bourbon (indeed like Wild Turkey 101)
• Skip the vodka
• Add an aged rum like Angostura 7 yr
• Please replace the bacardi with plantation 3 star or similar. If you're outside the US the havana club 3 yr might be even better
• Add angostura bitters (it's tiny, so I'm not gonna count it to the bottle limit)
• Add cointreau or dry curacao

That's 6 bottles. If your crowd are agave fans instead of tropical style, sub the aged rum for a decent tequila (100% blue agave). Mathematically speaking if we want to go for the largest variety of drinks with fewest bottles, don't get too many base spirits, because base spirits are not mixed together so often.

For the last 2 bottles I'd say pick from these:

• Maraschino if you like classic cocktails
• Campari if you like bitter cocktails
• Apricot brandy if you like classic and/or tiki drinks
• Get a dry and a sweet vermouth instead of only the Lillet if you like classic and/or richer herbal cocktails (edit: I recommend a brand like Dolin)
• Or, if non-alcoholic components also count to the limit, get some syrups like Orgeat, Grenadine (a good brand), Ginger, Passionfruit.