Quirky, non-rigorous, yet inspiring, literature in mathematics In contrast with such lucid, pedagogical, inspiring books such as Visualizing Complex Analysis by Needham and Introduction to Applied Mathematics by Strang, I've had the pleasure of coming across the non-rigorous, thought-provoking/stimulating, somewhat quirky works of Heaviside on operational calculus and divergent series; of Ramanujan on series, in particular, his use of his master theorem/formula (as explicated by Hardy); and the relatively unknown posthumous notes of Bernard Friedman on distributions and symbolic/operational calculus Lectures on Applications-Oriented Mathematics. 
These works just blindside you. You think, "What the hey?" and slowly they grow on you and you start to understand them after further research using other texts, translating the terminology/concepts, and working out details. You're left with a deeper understanding and appreciation of the originality and applicability of the work--much like Hardy professed the day after he received Ramanujan's letter, no doubt.
(Friedman's Applied Mathematics, in contrast, is of a very different nature and an immediately enlightening intro to its topics.)
Any similar experiences with other mathematical works?
(This question is not research-level per se and may be more appropriate for MSE, but certainly the works cited have inspired and continue to inspire much advanced research into the related topics, and the question falls in a similar category to MO-Q1 and the MO-Qs that pop up in the Related section of that question).
 A: Andrew Granville and Jennifer Granville's "Prime Suspects" is a comic book which despite being phrased as a murder mystery does a surprisingly lucid job connecting ideas involving primes with ideas involving permutations.  This paper Andrew Granville outlining the basic idea is pretty non-rigorous but fascinating. 
A: To elaborate on Piyush Grover's comment, large swaths of theoretical physics can be considered "non-rigorous yet inspiring."  The adjective "quirky" might not be so apposite, though.  But perhaps somewhat in the spirit of the question is Richard Feynman's semi-popular book QED: The Strange Theory of Light and Matter.  There are still some open questions about the mathematical consistency of QED.
A: This book is very elementary, but has a lot of excellent material.
@BOOK{Polya54,
  author =       {Polya, G.},
  title =        {Mathematics and Plausible Reasoning. Vol.1: Induction and Analogy in Mathematics. Vol 2. Patterns of Plausible Inference},
  publisher =    {Princeton University Press},
  year =         {1954},
}

A: I enjoyed reading the book 'Quantization, Classical and Quantum Field Theory and Theta Functions' by Andrej Tyurin very much. It is  certainly not rigorous but it was very inspiring for me.
