The verb say, when introducing indirect reported speech, takes as its complement a finite clause introduced by that. That much is clear. What is tricky is selecting a verb form which best fits the situation. The choices are:
- present tense form
- preterite form
- would + base form
As another respondent points out, the preterite form can be used to talk about something which was true at the reference time, but may or may not be true presently. The present tense form can only be used for something which was true at the reference time, and is still true now.
A key issue is that preterite forms of verbs in English have two main functions.
- Indicating past time
- Indicating modality
If you say:
I didn't think that you would say that you wanted to play baseball.
The preterite form want is being used to show irrealis modality (to discuss a situation which is in doubt or contrary to fact). On the other hand, if you say:
You said that you wanted to play baseball.
The preterite form of want is used to show past time.
What triggers the modal interpretation in the reported speech clause is the phrase I didn't think that... in the first matrix clause (the one whose main verb is think). This sets up the second matrix clause (the one whose inflected verb is would) to have a modal preterite auxiliary verb.
The sentence which has two would 's sounds a bit weird, firstly because there are two would 's in a row, and second because it is hard to decide whether the second would is being used (redundantly) for modality, or for tense matching.