How many Emperors did the Star Wars First Galactic Empire have?
Counting all versions and canon levels.
BLUF: There was one in Canon, possibly 4 in Legends.
If you visit the Galactic Empire Wookieepedia page, you'll see that this answer depends on whether you're talking about the Canon Empire or the Legacy Empire. The Legacy Empire could have arguably had 4 different emperors, one of which was a clone of Palpatine himself.
In Legends, there was originally Palpatine, obviously, but then after period of infighting, Grand Admiral Thrawn reunited the Empire and brought back an Imperial Resurgence. After Thrawn was defeated, a clone of the original Emperor came into power and even managed to put Luke Skywalker under his control for a time. Later, Carnor Jax assumed control for a time, but due to a lack of consolidation, had little power.
However, the Canon Empire only had one true Emperor. While the Empire was most certainly not completely defeated after the destruction of the second Death Star (The First Order is one surviving part of the Empire in Ep. VII), afterwards the Empire no longer had the reach or cohesion it had had under Palpatine. Palpatine's death marked a splintering of the Empire into different feuding factions who were fighting not only the Rebels and the New Republic, but themselves. As such, "Declarations of a new Emperor were announced almost every day, but none were able to consolidate any substantial power."
Some of the Admirals and Moffs loosely worked together against the Rebellion and later New Republic, but they never all worked together behind one ruler, only cooperated in the form of a council.
There is one Canon contender for being considered a second Emperor, self-proclaimed Counselor to the Empire Gallius Rax. Originally a Fleet Admiral and organizer of one of the Imperial Councils, Rax proclaimed this title and managed to unite the Empire for a short time before his defeat in the Battle of Jakku, after which there was the signing of the Galactic Concordance, which signaled the surrender of the Empire. After the battle, the Empire broke up into different ruling factions more or less at peace with one another, and have to date never fully reunited.
While Rax did reunite most of the empire, he did so only for a short time, and did not rule as much territory as Palpatine had, and was therefore not a true successor to the title of Emperor.
In this online copy of Star Wars IV: A New Hope George Lucas (ghost written by Alan Dean Foster) 1976, page 77, Luke asks Kenobie how his father died.
"He was betrayed and murdered," Kenobie declared solemnly, "by a very young Jedi named Darth Vader." Kenobie was not looking at Luke. "A boy I was training. One of my brightest disciples...one of my greatest failures."
Kenobie resumed his pacing. "Vader used the training I gave him and the force within him for evil, to help the later corrupt emperors. With the Jedi knights disbanded, disorganized, or dead, there were few to oppose Vader. Today they are all but extinct."
So Vader served "THE LATER CORRUPT EMPERORS". Thus there were at least two emperors after the first one by the time of Star Wars IV: A New Hope.
Since the prologue to the novelization tells how the ambitious Senator Palpatine became the first emperor, and since this passage shows that by the time of Star Wars IV: A New Hope there had been at least two successors to Palpatine the first emperor, I was shocked to read the emperor at the time of Star Wars VI: Return of the Jedi described as Emperor Palpatine in the novelization.
But I guess it could be explained by the Emperor in Star Wars VI: Return of the Jedi being Palpatine the Second (or higher), named after the first Emperor of the Galactic Empire.
The actual copy of Star Wars IV: A New Hope that was scanned to make the online preview quoted says Ballatine Books", "Del Rey", and "Lucas Books".
According to this news item from 2/21/00:
"Lucasfilm and Scholastic have teamed up to form an imprint named lucasbooks...".
The LucasBooks logo on the title page indicates the online copy ofStar Wars IV: A New Hope may have been printed in 2000 or later. So even after The Phantom Menace was written and produced, and released in 1999, Lucasfilm apparently did not bother to change anything in the Star Wars novelization.
So in so far as Star Wars novelizations count as canon, there is a big inconsistency in the number of galactic emperors in Star Wars.
The films may prove that the first and the last Galactic Emperors were the same person, but they don't prove that he was the emperor for the entire 24 year period.
It is possible that Palpatine was deposed by a usurper and later regained the throne.
Possibly the Galactic Emperor was supposed to be elective like the Queen of Naboo, so Palpatine could have "retired" and had a close associate under his influence and control elected for a term and "come out of retirement" and been elected to handle the emergency of the Rebel Alliance.
Porfirio Diaz (1830-1915) was dictator of Mexico from 1876-1911, but he was only President of Mexico from November to December 1876, from 1877 to 1880 and from 1884 to 1911. The President of Mexico from 1880 to 1884 was Manuel Gonzalez Flores, a close political ally of Diaz.
And Palpatine could have planned to alternate his terms as elected emperor with terms by his close political allies under his control and periodic crises engineered to have him called out of "retirement" when he desired. Possibly there was a term of "emperor Jar Jar Binks". Palpatine might have caused the formation of the Rebel Alliance in order to have a crises to recall him to the throne.
Of course this theory could be disproved by new Disney canon set during the period between Revenge of the Sith and A New Hope.