I'll convert this comment into an answer.
You can obtain this data rather clumsily with the Tooling API in the Developer Console. Enter the following in the Query Editor, and check the "Use Tooling API" box:
SELECT count(DeveloperName), TableEnumOrId FROM CustomField GROUP BY TableEnumOrId ORDER BY count(DeveloperName) DESC
That'll get you the objects with the highest count of custom fields at the top. Note that it supplies a mix of standard object names and custom object Ids, each one with the count of custom field attached to it. To find the name of a given Custom Object's Id, you can issue another query:
SELECT DeveloperName FROM CustomObject WHERE Id = 'THE_ID'
Or, of course, you can build yourself a translation table with
SELECT Id, DeveloperName FROM CustomObject