The ARRL hosts many contests throughout the year, and maintains a leaderboard of sorts to encourage competition.
Is it possible to rank on any of the leaderboards without knowing or using morse code?
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ARRL Hosts a list of their annual contests here.
As you can see from the list, there are a lot of CW based contents, but also Phone and RTTY.
I'd imagine that you wouldn't rank at all in an RTTY contest using CW, so it is possible to participate and rank without CW knowledge in at least some of the contests.
Also available here is a list of contest results sorted by year, you can view the results for Phone and other contests.
Most of the ARRL contests have separate events for phone and CW, November Sweepstakes, for example. Some contests have both modes, like Field Day; all the state QSO parties usually support all modes and bands. Non-ARRL contests run the full gamut of rules. The guys at the top of the heap are serious operators; they have their equipment and workflow finely tuned to maximize their score.
That doesn't mean you need a superstation, but if you don't you need to focus on the area you can excel at where the big guns don't qualify or participate.
So you can place on the leaderboards with just phone, but you can place on the CW leaderboards without knowing CW either. As you know, contest exchanges are highly scripted. So scripted you can use fldigi or some similar program to decode CW and send macros in CW (with the appropriate interface to your rig link a SignaLink).
Really, to search and pounce CW, you should be able to recognize you call as a backup to the decoder, but it's not necessary. Then, you only need two macros: 1) your call, 2) the exchange, for CQ WW: "5NN 5" (being in zone 5). That's it. Notice you never send the running station's call sign, you can do this all on macros. Now, strictly speaking, that's a weak excuse for a contact, but that's pretty much true of all contests anyways.