It is possible to not have to trim. I have a bike with a 6 speed block on back and double up front and I never have to trim the front. It's more by luck than design but it is a real pleasure to use. I prefer it to a brifter it's so fast. Sheldon indicated it is possible but probably only with select setups.

I have found that a slightly loose chainring bolt or two will be enough to cause chain rub on this bike so might be worth checking those.

Edit: Checked the cross chain combos (gears 6 and 7) and no rub there either. Chain is a fairly modern design - not sure what width.