If you want to take an example or several examples, you use the phrase 'for example,' not 'for examples.' Though the word 'example' is a countable noun, why is 'for examples' wrong?
When you say "for example" you're referring to the idea of "example". The "for" isn't a way to shorten "To further illustrate my point here is a list of examples". In this way "example" isn't countable.