I have two different business units that are wanting ways to verify that an object as well as certain related (not necessarily master-detail) objects have a specific number of attachments before allowing the main object to enter the approval process. In initial research, a team member found the project created by DouglasCAyers aka afawcett on GitHub named 'declarative-lookup-rollup-summaries', which can be found here: declarative-lookup-rollup-summaries. I am not sure that this fits our needs though. Attachments are apparently their own standard object and I am not sure that they fit the definition of a lookup field, although, I certainly could be very wrong about that.
Another solution that would definitely work, but involves a trigger was provided by bob_buzzard in August of 2011 here: Bob's answer. Basically, creating a trigger that selects a count from the attachment object based on the specific parentid. Tracking the decrement when an attachment is deleted would also be necessary.
Currently, both of these business units are using Salesforce in Classic, but we are planning to try and move our org to Lightning in 2018-2019. Since attachments in Lightning are not handled the same way as they are in Classic, I don't want to suggest a solution to our current problem that is going to make moving to Lightning any harder than if I had recommended/implemented a different solution. Also, I am open to any other thoughts and ideas beyond what I have already found.