It's easy to specify such a system that's more powerful than ZFC and has strong arguments for its consistency: ZFC+con(ZFC), which is consistent iff ZFC is consistent.

And it's easy to go one level up, I.e that system plus it's own consistency.

These systems will be stronger than ZFC, and will prove the machine in question halts.

Of course, the lowest unknowable from ZFC BB number is probably around BB(15), so likely none of those systems get that high, but they do probably give us more than just ZFC does.

See also http://www.scottaaronson.com/blog/?p=697 for some limitations on this.

Also, there are large cardinal axioms which imply the consistency of ZFC and are believed to be consistent, but I'm not so familiar with them. I think mathematicians would consider those axioms as "what such a system would look like".