Do we have any writings that are reasonably likely to be written by Apostles, but are not included in the Bible?
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One of the closest examples would be Paul's Letter to the Laodiceans. Scholarship is divided about the authorship, whether it is Paul or a Marcionite forgery - but the point is that at the time, when canonization was occurring, it was not accepted as such.
Most everything else (Gospel of Peter, Gospel of Thomas, etc...) was pretty much rejected even at the time as being inauthentic.
Oddly, Hebrews was often ascribed to Paul, but not universally. Still, that was its reason for inclusion.
At this point, even if we could authenticate a new work, that would not automatically make it scripture, however. Canonization was often supported by apostolic authorship, but it was cemented by common concurrence. At this point, the canon is closed, by consensus.
*P.S. I've walked through Laodecia, and seen the place on the picture. It's an amazing place!
There is another example: Epistle of Barnabas, which suffers from some of the same questions of date and authorship that the letter to the Laodiceans.