When I look at the most useful single request that I've done to agencies, it's requesting a FOIA log or a list of recent FOIA requests and replies. There's a few reasons why you might do this, and I'll illustrate with some recent results.
First, you get the very basic intelligence to know who is asking questions of the public body and how frequently those questions come up. Sometimes, ask in the case of the Ann Arbor Downtown District Authority, there are only a handful of FOIA requests over three months; on the other hand, a similar scoped request to the City of Ann Arbor returned hundreds of pages of documents. Sifting through these unearths names of reporters, what questions they are asking, and whether they follow up when they get rejected.
Second, the unanswered questions and the rejections can prompt new lines of inquiry. A out-of-town reporter asked the Ann Arbor Public Schools what the total cost of substitute teachers was for the district, and their query was rejected because the school system did not have an existing document with that figure on it. The follow up that's being pursued by a local schools blogger is to unpack the request into answerable parts and reassemble the answer, perhaps by looking for invoices or reports at a school by school level.
Finally, it's always good to keep tabs on what kinds of side conversations go on between reporters and FOIA officials. You'll see appeals and conflicts over fees, protests for delays, and questions of legality of various actions. Viewing FOIA requests gives you an insight into tradecraft of the reporting trade and can give citizen advocates insight into how to best phrase their questions and how to deal with obstacles as they come up.