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Runner on first. Batter rips one into RF. Runner rounds second and is headed to third, so right fielder fires one across to third base. Batter-runner advances to second on the throw. Throw is way high and to the fence - if it would have been a strike at the logo at the side of the bag, he probably would have gotten the runner out, but if it was a chest high, normal throw, there is no way the third baseman would have gotten the tag down in time. Runner gets up from his slide into third and takes off for home. Batter-runner, who has taken a wide turn at second, heads to third when he sees the error and no one covering. Third baseman recovers the ball and fires home, striking the runner in the back as he slides into home. Ball squirts toward the mound, catcher recovers and cocks like he's going to throw to third, but the batter-runner completes his run to third without further attempt at a play.

Obviously there was an error on the right fielder that permitted the runner and the batter-runner to each advance a base beyond where they would have been had the wild throw not occurred.

We have two scorers for the team - a dad that keeps the paper book (and thus "official") and myself on iScore to quickly compile stats and send box scores to local media. On close or questionable plays, we regularly consult so that we have the same things down.

He says it's a no brainer that it's two errors because the ball struck the runner - always an error. My argument for only one error is:
the play at the plate:
1.) was too close to say was an obvious out,
2.) was already accounted for in the throwing error by the right fielder.

The batter-runner's advance was also covered by the throwing error by the right fielder. Had the throw to the plate not hit the runner, the return throw to third might have caught the batter-runner, might not. It all depends on how much time the catcher would have taken to block the plate, make the tag, and return the throw.

So, am I being too generous in wanting to score only a single error to the right fielder?
(Oh, and to help you with the "ordinary effort" clause in the rule covering errors, this is varsity high school level ball.)

In the perfect world....runner advancing to 3rd is out....but, as you wrote...he was most likely going to be safe...so I would score it as a single, with the runner advancing to third by the batter and then the runner advancing to second on the throw....
Then I would score the runner advancing home on the error by the RF (he does not advance to home without the error on the RF), with the batter advancing to 3B on the new error by the 3B. (he does not advance to 3B without the new error)

Total judgement call though....I believe that based on your description, it could be scored either way...

I think this is one error only.

Single error only in my opinion. The ball striking the runner is *NOT* always an error. In fact, the overthrow from right isn't always an error. If 3B had recovered the ball and gotten the out at home, then there would be no error on the play at all.