I asked this question about two weeks ago on MSE and haven't gotten an answer, so I thought I would post the question here.

Do there exists sentences which are independent of **ZFC**, cannot be shown to be independent through some method of forcing, and do not increase the consistency strength of **ZFC** (e.g. so Large Cardinal Axioms are out)?

If there does exist such a sentence, I would love to know a concrete example. One with a combinatorial flavor would be ideal.

Edit: Feel free to increase the consistency strength of **ZFC** (say **ZFC+ $\Delta$**) in the "meta-"sense (i.e. work where you want to work). Does there exist a "non-forcible" independent sentence that does not increase the consistency strength of **ZFC**?