For almost the entire war, Hermann Göring was Hitler's designated successor. He had been the second most powerful man in the Reich since the Nazi takeover, and had done much of the practical work of leading the country.
At the outbreak of war with Poland, Hitler had announced in a speech that Göring would succeed him "if anything should befall me." This was emphasised by Göring's promotion to Reichsmarshall in 1940, making him the highest-ranking officer in Germany, second only to the supreme commander, Hitler. In June 1941, Hitler issued a formal, although secret, decree, naming Göring his successor and deputy in all his roles. In particular, Göring had the right and duty to act in Hitler's place if Hitler were incapacitated.
By 1943, Göring was falling out of favour, although the decree remained in place. This was the point at which Speer started being talked of as a possible successor. Hitler could have made him the successor with a spoken sentence. He didn't have to deal with a constitution, legal system or legislature for this. He could do whatever he wanted. However, he did nothing.
In April 1945, after Göring had left Berlin for Berchtesgaden, and the Soviets had entered Berlin, Hitler suggested that Göring might be better suited to negotiate peace terms. The Luftwaffe Chief of Staff, Karl Koller, who was loyal to Göring, left Berlin and went to tell Göring. He, Koller and Hans Lammers, the secretary of the Reich Chancellery, concluded on examining the 1941 decree that Hitler was, or very soon would be, incapacitated, since he would be captured, killed or lose communications, and it was therefore Göring's duty to take over.
Göring sent Hitler a telegram asking for confirmation of this. Martin Bormann, Hitler's secretary, spun this into an attempt to unseat Hitler. Hitler was furious, and sent a telegram back telling Göring that he had committed high treason, and that he should resign all his offices immediately, or he would be shot.
Four days later, Hitler wrote his will, designating Goebbels as Chancellor and Doenitz as President, separating the two main offices that Hitler had held. He also sacked Göring from all his offices and expelled him from the Nazi Party. Hitler killed himself the next day. So while Goebbels and Doenitz were Hitler's designated successors at the time of his death, they'd held those positions for about a day before succeeding. Goebbels, of course, killed himself the next day. His wife did the same, after killing their children.
It's a serious mistake to look for a lengthy list of people in the order of succession, like that of the USA. Hitler's policy was that senior leaders were in competition with each other, and the constitution was "Whatever the Führer says it is." Thinking of the senior Nazis as the quarrelling heads of clans within a barbarian tribe is a much better model than civilised politicians in an advanced state.
Addendum: Goebbels had a de facto successor as Chancellor, Lutz Graf Schwerin von Krosigk, who had been finance minister since 1932. Doenitz had asked him to form a new administration, and he did. It's known as the Flensburg Government, and von Krosigk used the title of "Leading Minister" rather than Chancellor.
Sources: the Wikipedia page linked above, as a reminder of the dates, The Bunker, by James P O'Donnell, and the German official history, Germany and the Second World War, volumes V/IIA and V/IIB.