The modern sense, applying chiefly to numbers and physical extensions, dates from the 14th century, and this will be found in Shakespeare. But there are other nuances which are nothing to do with "adding up" at all.
When Hamlet complains about the way the Danes "with swinish phrase Soil our addition" (Hamlet, I.iv.20), he doesn't mean they were criticising their arithmetic. Here, addition means simply "title, name". The usage is closely related to one in heraldry, where an addition is a mark of honour, often added to a coat of arms. When Ajax tells Hector, after fighting with him, "I came to ... bear hence A great addition earned in thy death"
(Troilus and Cressida, IV.v.141), he means no more than "distinction". And it is this concept of "something extra" which can be detected in several other uses. It means "external honours" when Lear, after dividing his kingdom among his daughters, wants still to "retain The name and all th'addition to a king" (King Lear, I.i.136). And it means "exaggeration"
when the Captain tells Hamlet "Truly to speak, and with no addition"