What do they like?
-Visit one or more of the harbor islands
-Head out to Springfield and visit the Basketball Hall of Fame
-Day trip to Kittery for outlet shopping
-Explore Cape Ann (even if they're not beach kids, the singing sand might interest them)
-Trip out to the Berkshires- Tanglewood is questionable with teens (some may like it, some may not), some of the ski resorts have summer amusements- alpine slides, mountain coasters, etc.
-Water Country (Portsmouth, NH) has discounts for weekday tickets
-Walk around Salem
-Scavenger hunt- there are a few companies that offer fairly inexpensive hunts done by clues sent to your phone. I've seen a few on Groupon before. Urban Adventure Quest is one, and I'm blanking on the other... there's one that offers different kinds of hunts.
-Head down to Providence for a day- check when they're having a Water Fire event, words cannot even describe that. So cool, and lots of free giveaways by the sponsoring organizations.
-Also Providence- Roger Williams Park Zoo. They sometimes have days when you can pay a canned good for admission.
-Do you and the kids bike? Go to the Cape Cod Canal for a day, or head over the bridge to Falmouth for the Shining Sea Bikeway, or even further for the CC Rail Trail (I recommend Orleans, with a stop at Chocolate Sparrow for the world's best frozen hot chocolate)
You said no money drains/Apex, but if they haven't tried it yet and there's some money for one special activity, the virtual reality at the Apex Center is really amazing. It's certainly not cheap, but as a one-time thing if there are some extra funds floating around, it's fun.
Also pricey but might be possible if you find discounts (do they still do the Coke cans?) would be Six Flags in Agawam or Canobie Lake in NH. When I was a teenager (back when Six Flags was Riverside), our favorite day trip was actually to Water Country in Portsmouth, NH.
Edited to add: Also, check your local YMCA (even if it's in a different town)- mine is offering a variety of classes and workshops for children and teens this summer for low prices or even for free.