There are many awesome examples out there where you can get a very direct feeling for what Machine Learning is. I'll collect a couple of them here.
Image Input Data ¶
Math Symbol Recognition ¶
The write-math.com web service allows you to recognize mathematical symbols automatically. It is described in my bachelors thesis. The HWRT data is available, also in its rendered form as HASYv2 dataset.

Drawings ¶
Quickdraw lets you recognize drawings. They built a database of 50 million drawings.

Using this kind of data, you can create an application which recognizes what was drawn and improves the drawing. autodraw.com does so:
Deep Dream ¶
Deep Dream is a technique which needs a few more words to explain. You can read my paper Creativity in Machine Learning if you are interested in a very high level overview. If you just want to play with it, have a look at deepdreamgenerator.com.

Style Transfer ¶
You have two images: A style image (e.g. by a painter) and a source image. You want the source image to be in the style of the style image:
Source image:

Style image:

Output image:

https://deepart.io seems to be a web service for this kind of machine learning. I didn't try it, though.
Super-Resolution ¶
You have a small image and want the same image, but with higher resultion? Search for "super resolution":
Text Input Data ¶
Translation ¶
DeepL.com is much better than https://translate.google.com, but also more restricted. As always, you can find some... interesting... translations:

Sentiment Analysis ¶

Text To Speech (TTS) ¶
Lyrebird.ai has the most impressive TTS system I have seen so far (although Googles Tacotron 2 audio samples are impressive as well).
Other ML ¶
Clustering ¶
Besides my small k-means clustering example, there is Tensorflow Projector
2D Embeddings ¶
Mapping datapoints in 2D makes it easier to find what you are looking for. Have a look at fontmap:

Classifiers ¶
Karpathy made a couple of interactive examples which show the decision boundaries of classifiers:

Honorable Mentions ¶
- howhot.io was a service which lets you upload an image with a face and rate how hot the person is. It is no longer available.
- Microsoft Chatbot Tay went racist (source)
- A visual introduction to machine learning
- Martin Thoma, 2016: Creativity in Machine Learning
- Alex Rogozhnikov, 2016: Gradient Boosting explained
- Alex Rogozhnikov, 2016: Hamiltonian Monte Carlo explained
- GridWorld: Dynamic Programming Demo
- Wattenberg et al, 2016: How to Use t-SNE Effectively