[The following is an alternate timeline that I first posted to the soc.history.what-if newsgroup on January 22, 2001. The subject matter was suggested by my wife. I re-post it here to preserve it for posterity.]
20 April 1889
Klara Pölzl Hitler is shocked and dismayed to learn that her newborn child Adolf has accidentally drowned while being washed by the midwife. Klara has now lost a total of four children in infancy. She suffers a mental breakdown, becomes terminally depressed, and dies a month after baby Adolf.
12 September 1919
Anton Drexler, founder of the German Workers Party, is sitting at a party meeting, listening to one speaker after another drone on and on. What the party needs, he figures, is someone with the gift of gab, someone they can use to mobilize mass support. Drexler looks around the room at the two dozen or so party members he has been able to attract, and sighs. While he's wishing, he might as well wish for the moon. He's just as likely to get it.
2 February 1932
Heinrich Bruening, leader of Germany's third-largest political party, the Catholic Center, meets with an ambitious army officer named Kurt von Schleicher. Schleicher has a proposal for Bruening. Everyone, Schleicher explains, expects Chancellor Alfred Hugenberg, leader of the German National Party, to be elected President in the upcoming election on 13 March. However, Schleicher has learned from a high-ranking member of the German National Party named Ernst Röhm that there is growing dissatisfaction with Hugenberg within the radical wing of the party. Gregor Strasser, the leading figure among the radicals, would like to depose Hugenberg as party leader, but knows he doesn't have enough support. Schleicher would like to arrange a meeting between Strasser and Bruening to discuss the possibility of Strasser leading the radicals out of the German National Party and into the Catholic Center Party. With their support, Bruening will be able to win the Presidency, and Strasser can become Chancellor of a new Catholic Center government.
It sounds feasible, Bruening says. Go ahead and arrange the meeting.
Schleicher leaves Bruening with a spring in his step. He has no intention of letting Gregor Strasser become Chancellor of Germany, but there's no need to let either Strasser or Bruening know that at this point. Schleicher has already arranged for separate meetings with Röhm and with Strasser's deputy Joseph Goebbels. There are many webs yet to be spun today.
8 May 1932
Alfred Hugenberg, newly-elected President of the German Republic, is uneasy about his upcoming meeting with Kurt von Schleicher. He has heard any number of rumors about Schleicher's role in the bizarre series of betrayals and double-crosses that attended his election. The plans he had for the composition of his government are in disarray. Strasser is in disgrace, Ernst Röhm now heads a breakaway faction of the German National Party, and Joseph Goebbels keeps urging him to arrange a coalition government with Heinrich Bruening's Catholic Center Party.
When Schleicher arrives, he is accompanied by another man. Herr President, says Schleicher, I would like to introduce you to your next Chancellor. His name is Franz von Papen....
13 August 1932
President Alfred Hugenberg stares out of the window of his office. From where he is standing, he can see at least half a dozen plumes of smoke rising from burning buildings. Behind him he hears the voice of his Chancellor, Kurt von Schleicher.
Herr President, says Schleicher, we must act now. Germany is in chaos, Röhm's bully boys have turned Berlin into a battleground, and the Communists are on the verge of open revolution. You have no choice. You must invoke Article 48 of the Constitution and grant the government dictatorial powers. It is the only way to take control of the situation.
President Hugenberg continues to face the window. Although the idea fills him with foreboding, he fears that Schleicher is right. He has no choice.
8 October 1932
Ernst Röhm, Führer of the New National Party, nods in satisfaction as his men drag the bodies of Alfred Hugenberg and Kurt von Schleicher out of the President's office. From outside the shattered window, he can hear his men chanting, Haut'se doch zusammen, haut'se doch zusammen! Diese gotverdammte Juden Republik! Let's smash it up, let's smash it up! That goddammed Jew republic!
Now, thought Röhm, we can cleanse the Fatherland of the Jews that stain it.
30 June 1937
Marshal Jósef Piłsudski walks through the burned-out shell of the Reichstag building. He is not looking forward to the upcoming meeting with his French and British counterparts at Potsdam. They would be urging lenience in dealing with the defeated Germans, and he is in no mood to listen. His troops, many of them Jewish, have brought back photographs from the concentration camps. Despite all he has seen in the course of an eventful life, Piłsudski has been sickened. Perhaps he could arrange tours of the camps for the British and French delegations. That might make them see reason.