Although the only explicitly named Nazgûl is Khamûl, there is strong evidence for another: Gothmog, Lieutenant of Morgul.
Gothmog is only named in one place in all of Tolkien's writings, in the Return of the King chapter "The Battle of the Pelennor Fields":
There they had been mustered for the sack of the City and the rape of Gondor, waiting on the call of their Captain. He now was destroyed; but Gothmog the lieutenant of Morgul had flung them into the fray...
This straight away establishes a link between the Witch-king and Gothmog, and if we look at the etymology of the word "lieutenant" we see that it comes from lieu, "in place of", and tenant, "holding", effectively meaning a second-in-command who takes full command when his lord is absent.
Tolkien uses the word "lieutenant" in several other places in his writings, and as a philologist he would be well aware of this meaning, and his use of it establishes this awareness. Therefore Sauron is described as the lieutenant of Morgoth, the Stewards of Gondor are described as lieutenants of the Kings, and so on.
Now, I am aware of the depiction in Peter Jackson's movies, but this depiction is not based on anything that Tolkien actually wrote: Tolkien never explicitly says which species Gothmog belongs to.
I am also aware that the Wikipedia article on Gothmog discusses and discounts the possibility of his being a Nazgûl on the grounds that Khamûl was named in Unfinished Tales as their second-in-command. However, and a crucial flaw in this argument: Khamûl was in command of Dol Guldur, and not at Minas Morgul, following Sauron's return to Mordor (Unfinished Tales, "The Hunt for the Ring"):
Now at that time the Chieftain of the Ringwraiths dwelt in Minas Morgul with six companions, while the second to the Chief, Khamûl the Shadow of the East, abode in Dol Guldur as Sauron's lieutenant, with one other as his messenger.
Here we also see the same use of the word "lieutenant" I describe above.
I am finally also aware that the Mouth of Sauron is described as "The Lieutenant of the Tower of Barad-dûr" but it's also established that none of the Nazgûl were based at Barad-dûr so this shouldn't be seen as an objection.
None of this discounts the possibility that Gothmog was something other than a Nazgûl but it seems extremely improbable that the "Lieutenant of Morgul" would be a mere Orc.
By this reasoning there are therefore three named Nazgûl:
- The Witch-king of Angmar, the Black Captain, Lord of Morgul.
- Khamûl, Shadow of the East, the Black Easterling, Lieutenant of Sauron at Dol Guldur.
- Gothmog, Lieutenant of Morgul.
There is no evidence I am aware of for any other named Nazgûl.