Brain
Experimenting with intelligence
N-gram Model derived from scratch
Basic early models for predicting next characters based on statistics.
Deriving from scratch
Learning the inner workings of feed forward and back propagation.

Image segmentation
K-means clustering on an image

Universal function approximator
Learning Taylor Series with Backprop

Convolutions in Image Processing
Live image processing with different kernels

Optimization Functions
Surfaces like the Ackley or Bukin functions used as benchmark for optimization problems

DBSCAN Algorithm
Density-based, noise redundant clustering algorithm.