I'm not a practitioner, so read with some skepticism, but here's my list:

* Calculus

* Real Analysis

* Statistical Mechanics

* Probability

I'm not sure I have any good recommendations for Calculus, but for real analysis, I would recommend "The Way of Analysis" by Strichartz [0].

I don't have good recommendations for books on statistical mechanics, as I haven't found a book that isn't entrenched in coming from a physics perspective and teaches the underlying methods and algorithms. The best I can recommend is "Complexity and Criticality" by Christensen and Moloney [1], but it's pretty far afield of statistical mechanics and the like. Simulating percolation, the Ising model and ricepiles uses a lot of the same methods as financial simulation (MCMC, etc.).

For probability, I would recommend "Probability and Computing" by Mitzenmacher and Upfal [2], "Probability ..." by Durrett [3] and Feller Vol. 1 and 2 [4] [5] for reference.

I also would recommend "Frequently asked questions in Quantitative Finance" by Wilmott [6].

Also know that there's a quantitative finance SO [7] that might be helpful.

[0] https://www.amazon.com/Analysis-Revised-Jones-Bartlett-Mathe...

[1] https://www.amazon.com/COMPLEXITY-CRITICALITY-Imperial-Colle...

[2] https://www.amazon.com/Probability-Computing-Randomization-P...

[3] https://www.amazon.com/Probability-Theory-Examples-Durrett-H...

[4] https://www.amazon.com/Introduction-Probability-Theory-Appli...

[5] https://www.amazon.com/Introduction-Probability-Theory-Appli...

[6] https://www.amazon.com/Frequently-Asked-Questions-Quantitati...

[7] https://quant.stackexchange.com/

Anything that would help build a strong probability theory background.

Assuming you have some freshman-college math literacy, here are a few great introductory books to Probability theory:

    Introduction To Probability by D. Bertsekas & J. Tsitsiklis 

https://www.amazon.com/Introduction-Probability-2nd-Dimitri-...

    Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance  by K. L. Chung & F. AitSahlia  

https://www.amazon.com/Elementary-Probability-Theory-Introdu...

    An Introduction to Probability Theory and Its Applications Vol. 1 by W. Feller 

https://www.amazon.com/Introduction-Probability-Theory-Appli...

I'm sure there are some great video lectures on the subject as well, but unfortunately I can't point you to any relevant material since textbooks is what I used when I was in college. But if I had to guess, the latest relevant MIT OCW course that has video lectures available would be sufficient.

Feller's two volume classic [1] has plenty of motivation - his exposition of combinatorics at the beginning of the first volume is a great introduction to that subject! However, it wasn't written with algorithms in mind.

Venkatesh's more recent volume [2] is very well motivated and is better suited to modern engineering applications. Both have lots of exercises.

[1] https://www.amazon.com/Introduction-Probability-Theory-Appli...

[2] https://www.amazon.com/Theory-Probability-Explorations-Appli...