In no particular order:

• Generalized Additive Models in R: A Free Interactive Course, by me. A friendly introduction requiring only basic knowledge of R and linear regression. 4-5 hours of slides and interactive exercises. This was formerly on a commercial platform but is now open source.
• Slides from my talk at the New York R User's Group (this repo). This is a high-level overview of things that GAMs and mgcv can do. Video (~80 minutes) here: https://www.youtube.com/watch?v=q4_t8jXcQgc
• Materials from our 2017 Ecological Society of America workshop on GAMs. These are designed for the interactive workshop but may still be useful. Target audience is graduate students with a little more statistical training. GLM knowledge a prerequisite. More material on inference, theory and more exotic distributions. See the references page in particular
  • The 2018 materials are almost exactly the same, but we are tracking issues in that repo for future improvement!
• Gavin Simpson's ~3 hour YouTube introduction to GAMs covers much of the same material as my course in one long lecture, with different examples and some updates as of summer 2020.
• The essential GAMs reference is Simon Wood's Generalized Additive Models in R.
  • Recently reviewed by Virgilio Gomez-Rubio in the Journal of Statistical Software
• An online book by Michael Clark gives an a very nice short introduction to both GAM theory and use in R.
• Gavin Simpsons's excellent, GAM-centric blog where he tries out new and little-used GAM formulations.
• StackOverflow and Cross Validated tags for mgcv
  • Home to Gavin's amazing Cross Validated answer on spatiotemporal modelling with GAMs
• A post by Kim Larsen's GAMs on the Stitchfix Blog which explains GAMs and compares them to other methods for classification.
• The gratia package by Gavin Simpson for using mgcv with ggplot2 and other useful and tidy helper functions, such as calculating spline derivatives and simulating from model posteriors. Here's a blog post introducing it.
• Hierarchical Generalized Additive Models: an introduction with mgcv A paper by Eric J. Pedersen, David L. Miller, Gavin Simpson, and Noam Ross on fitting gams with heirarchical/mixed structures. GitHub repo here.
• Modelling palaeoecological time series using generalized additive models, a paper by Gavin L. Simpson
• Bayesian views of generalized additive modelling, a brief but useful write-up on Bayesian approaches and interpretations of GAMs by Dave Miller.
• Generalised additive mixed models for dynamic analysis in linguistics: a practical introduction by Márton Sóskuthy
• Simplified Integrated Nested Laplace Approximation by Simon N. Wood, details the ginla() function added in mgcv 1.8-27.