I was asked for a source on this today. I couldn't find this on hexbear anywhere and other places that do have it have been slowly disappearing or becoming harder to find over time. So I thought I'd preserve it here.

EDIT: Alaskaball has also made a series of comments with excellent book quotes referencing some of these attempts: https://hexbear.net/comment/5288072

May 1924, 23-31 (Marxist Internet Archive, "The Trotskyist Opposition Before and Now") ( https://www.marxists.org/reference/archive/stalin/works/1927/10/23.htm#1)

It is said that in that "will" [Lenin's Testament - ZB] Comrade Lenin suggested to the congress that in view of Stalin's "rudeness" it should consider the question of putting another comrade in Stalin's place as General Secretary. That is quite true. Yes, comrades, I am rude to those who grossly and perfidiously wreck and split the Party. I have never concealed this and do not conceal it now. Perhaps some mildness is needed in the treatment of splitters, but I am a bad hand at that. At the very first meeting of the plenum of the Central Committee after the Thirteenth Congress [Undefined date of this attempt, however, within the Thirteenth Congress and thus anywhere within the 23rd to the 31st - ZB] I asked the plenum of the Central Committee to release me from my duties as General Secretary. The congress itself discussed this question. It was discussed by each delegation separately, and all the delegations unanimously, including Trotsky, Kamenev and Zinoviev, obliged Stalin to remain at his post.

What could I do? Desert my post? That is not in my nature; I have never deserted any post, and I have no right to do so, for that would be desertion. As I have already said before, I am not a free agent, and when the Party imposes an obligation upon me, I must obey.

A year later I again put in a request to the plenum to release me, but I was again obliged to remain at my post.

What else could I do?

August 19, 1924 (Grover Furr, Khrushchev Lied, p. 244):

To the Plenum of the CC [Central Committee] RCP [Russian Communist Party]

One and a half years of working in the Politburo with comrades Zinoviev and Kamanev after the retirement and then the death of Lenin have made perfectly clear to me the impossibility of honest, sincere political work with these comrades within the framework of one small collective. In view of which, I request to be considered as having resigned from the Pol[itcal] Buro of the CC.

I request a medical leave for about two months.

At the expiration of this period I request to be sent to Turukhansk region or to the Iakutsk oblast', or to somewhere abroad in any kind of work that will attract little attention.

I would ask the Plenum to decide all these questions in my absence and without explanations from my side, because I consider it harmful for our work to give explanations aside from those remarks that I have already made in the first paragraph of this letter.

I would ask comrade Kuibyshev to distribute copies of this letter to the members of the CC.

With com[munist] greet[ings], J. Stalin.

December 27, 1926 (Grover Furr, Khrushchev Lied, p. 244):

To the Plenum of the CC [Central Committee] (to comrade Rykov). I ask that I be relieved of the post of GenSec [General Secretary] of the CC. I declare that I can work no longer in this position, I do not have the strength to work any more in this position. J. Stalin.

December 19, 1927 (Grover Furr, Khrushchev Lied, p. 245) (https://livrozilla.com/doc/796199/pelo-socialismo):

Stalin: Comrades! For three years [Suggesting there could be more resignation attempts unbeknownst to me - ZB] I have been asking the CC [Central Committee] to free me from the obligations of General Secretary of the CC. Each time the Plenum has refused me. I admit that until recently conditions did not exist such that the Party had need of me in this post as a person more or less severe, one who acted as a certain kind of antidote to the dangers posed by the Opposition. I admit that this necessity existed, despite comrade Lenin's well-known letter [Lenin's Testament - ZB], to keep me at the post of General Secretary. But these conditions exist no longer. They have vanished, since the Opposition is now smashed. It seems that the Opposition has never before suffered such a defeat since they have not only been smashed, but have been expelled from the Party. It follows that now no bases exist any longer that could be considered correct when the Plenum refused to honor my request and free me of the duties of General Secretary. Meanwhile you have comrade Lenin's directive which we are obliged to consider and which, in my opinion, it is necessary to put into effect. I admit that the Party was compelled to disregard this directive until recently, compelled by well-known conditions of inter-Party development. But I repeat that these conditions have now vanished and it is time, in my view, to take comrade Lenin's directive to the leadership. Therefore I request the Plenum to free me of the post of General Secretary of the Central Committee. I assure you, comrades, that the Party can only gain from doing this.

Dogadov: Vote without discussion.

Vorshilov: I propose we reject the announcement we just heard.

Rykov: We will vote without discsussion...We vote now on Stalin's proposal that he be freed from the General Secretaryship. Who is for this proposal? Who is against? Who abstains? One.

The proposal of comrade Stalin is rejected with one abstention.

Stalin: Then I introduce another proposal. Perhaps the CC [Central Committee] will consider it expedient to abolish the position of General Secretary. In our Party's history there have been times when no such post existed.

Voroshilov: We had Lenin with us then.

Stalin: We had no post of General Secretary before the 10th Congress.

Voice: Until the 11th Congress.

Stalin: Yes, it seems that until the 11th Congress we did not have this position. That was before Lenin stopped working. If Lenin concluded that it was necessary to put forward the question of founding the position of General Secretary, then I assume he was prompted by the special circumstances that appeared with us before the 10th Congress, when a more or less strong, well-organized Opposition within the Party was founded. But now we proceed to the abolition of this position. Many people associate a conception of some kind of special rights of the General Secretary with this position. I must say from my experience, and comrades will confirm this, that there ought not to be any special rights distinguishing the General Secretary from the rights of other members of the Secretariat.

Voice: And the duties?

Stalin: And there are no more duties than other members of the Secretariat have. I see it this way; There's the Politburo, the highest organ of the CC; there's the Secretariat, the executive organ consisting of five persons, and all these five members of the Secretariat are equal. That's the way the work has been carried out in practice, and the General Secretary has not had any special rights or obligations. The result, therefore, is that the position of General Secretary, in the sense of special rights, has never existed with us in practice, there has been only a collegium called the Secretariat of the CC. I do not know why we need to keep this dead position any longer. I don't even mention the fact that this position, called General Secretary, has occasioned in some places a series of distortions. At the same time that at the top no special rights or duties are associated with the position of General Secretary, in some places there have been some distortions, and in all the oblasts there is now a struggle over that position among comrades who call themselves secretaries, for example, in the national CCs. Quite a few General Secretaries have developed, and with them in the localities special rights have been associated. Why is this necessary?

Shmidt: We can dismiss them in the localities.

Stalin: I think the Party would benefit if we did away with the post of General Secretary, and that would give me the chance to be free from this post. This would be all the easier to do since according to the Party's constitution there is no post of General Secretary.

Rykov: I propose not to give comrade Stalin the possibility of being free from this position. As concerns the General Secretaries in the oblast and local organs, that should be changed, but without changing the situation in the CC. The position of General Secretary was created by the proposal of Vladimir Il'ich. In all the time since, during Vladimir Il'ich's life and since, this position has justified itself politically and completely in both the organizational and political sense. In the creation of this organ and in naming comrade Stalin to the post of General Secretary the whole Opposition also took part, all those whom we have now expelled from the Party. That is how completely without doubt it was for everyone in the Party (whether the position of General Secretary was needed and who should be the General Secretary). By which has been exhausted, in my opinion, both the question of the "testament" (for that point has been decided) and exhausted by the Opposition at the same time just as it has been decided by us as well. The whole Party knows this. What has changed now after the 15th Congress and why is it necessary to set aside the position of General Secretary.

Stalin: The Opposition has been smashed.

(A long discussion followed, after which:)

Voices: Correct! Vote!

Rykov: There is a proposal to vote.

Voices: Yes, yes!

Rykov: We are voting. Who is for comrade Stalin's proposal to abolish the post of General Secretary? Who is opposed? Who abstains? Noone.

October 16, 1952 (http://soviethistory.msu.edu/1954-2/succession-to-stalin/succession-to-stalin-texts/stalin-on-enlarging-the-central-committee/):

This article was taken from the Russian newspaper Glasnost devoted to the 120th Anniversary of Stalin’s birth, was the last speech at the CC [Central Committee] CPSU [Communist Party of the Soviet Union] before Stalin died. The text was being published for the very first time in the Soviet Union...

...MOLOTOV – [Glasnost -] coming to the speaker’s tribune completely admits his mistakes before the CC, but he stated that he is and will always be a faithful disciple of Stalin.

STALIN – (interrupting Molotov) This is nonsense. I have no students at all. We are all students of the great Lenin.

[Glasnost -] Stalin suggested that they continue the agenda point by point and elect comrades into different committees of state.

With no Politburo, there is now elected a Presidium of the CC CPSU in the enlarged CC and in the Secretariat of the CC CPSU altogether 36 members.

In the new list of those elected are all members of the old Politbiuro – except that of comrade A. A. Andreev who, as everyone knows now is unfortunately completely deaf and thus can not function.

VOICE FROM THE FLOOR – We need to elect comrade Stalin as the General Secretary of the CC CPSU and Chairman of the Council of Ministers of the USSR.

STALIN – No! I am asking that you relieve me of the two posts!

MALENKOV – coming to the tribune: Comrades! We should all unanimously ask comrade Stalin, our leader and our teacher, to be again the General Secretary of the CC CPSU.

Same attempt (A. I. Mgeladze, Stalin. Kakim ia ego znal. Strannitsy nedavnogo poshlogo. p. 118):

At the first Plenum of the CC [Central Committee] of the CPSU [Communist Party of the Soviet Union] called after the XIX Congress of the Party (I had been elected member of the CC and took part in the work of this Plenum), Stalin really did present the question of General Secretary of the CC CPSU, or of the post of Chairman of the Council of Ministers of the USSR. He referred to his age, overwork, said that other cadres had cropped up and there were people to replace him, for example, N.I. Bulganin could be appointed as Chairman of the Council of Ministers, but the CC members did not grant his request, all insisted that comrade Stalin remain at both positions.