We're Agile now, we don't do testing any more!

Scrum only has Product Owner, Scrum Master and Developer.
Testing is not part of the Scrum process, so we can eliminate it - and save money by sacking the testers.

I've had the joy of working with a company that seriously thought about taking this road.
It was a shock for me not because I come from a testing background, but because it's wrong on so many levels:

Agile means: Deliver valuable software

Buggy software isn't valuable. We don't expect to get praised for a "job well done" if the result isn't both usable and useful. How do you know that it's usable without testing? How do you know it's useful without testing?`

Agile means: Deliver value often

Working on bugfixes drains resources. The only way to prevent bugfixes is by not introducing bugs to the customer in the first place. How do you know that you don't have bugs without testing?

Agile means: Deliver value fast

You need a profound understanding of what your components do - and why. Without testing, you will be spending more and more time in understanding the impact of change as the project progresses. Specifically, without systematic testing, you may completely lose control!

Agile means: Eliminate waste

A standard trumpet: Testing is waste, because if there are no defects, tests don't add any value.
Well. That may sound true if you have a superficial understanding of (software) engineering. 

Ask yourself: Would you hire an architect who didn't run calculations to verify that your house won't crash over your head? No. Would you ride a car where designers didn't validate traffic safety regulations? No. 

So why do you want to build a piece of software without those checks? 

Because bankruptcy isn't as bad as death? Great! Tell me your company name and I will make a fortune by short-selling!

Agile means: Working with feedback

Each time a piece of code is finished, it should be exposed to customer feedback. Unless you want to look like a fool in front of your paying customers, you better have a strategy in place to make sure that they like what they see. You must verify the correctness and validate the applicability of your solution to their problem before confronting them with results. And the formal process for doing them is called "testing".

Conclusion

Not testing is planning for commercial suicide.

It holds true for Agile even more than for traditional projects, because in a Waterfall, you might get the customer to sign off some crappy product just for political reasons. In Agile, your customer will know that you don't know your trade within a couple weeks.