Why do we have VAT on (most) services? My knowledge of economics is quite limited, but as far as I understand, the "added value" which is taxed by value-added tax refers to the difference in the cost of production and the cost in the store. Since the product has to be processed, packaged, transported etc. the cost will rise because every person in this chain wants a bit of money for themselves. The government then taxes this added value. Correct?

If this is true, then why is there VAT on services? Most of the services are "supplied" directly to the customer. For example, if a hairdresser cuts your hair, there were no intermediate persons in this process and the cost of the hair cutting service didn't rise because of them. If you're a web designer, you make a website and provide it directly to the customer - it doesn't have to be transported etc.

Basically, why would someone have to pay value-added tax if there is no added value?

(This also goes for all goods that are sold directly to the customer - but unlike services, goods can be passed on (the customer can sell the good that he has bought) whereas services can't, so I think that most services should be exempt from VAT).
Services are just as much VATable supplies as pyhsical products are

It is a mistake to think of the increase in price along the supply chain as just each person wanting their cut. For them to achieve this they need to be increasing the value that can be delivered to end customers, even if - for example - it is simply bringing a pallet of product to a convenient location and being there so that customers can buy single items at a time and place that suits them. That adds value to the pallet of goods in a warehouse.

The value added is the differemce between the VATable outputs I can make and the VATable inputs I use. 

If my  business uses a consultant's time I pay input VAT on the money they charge me. I can offset this against the VAT I had to collect from my customers and send to HMRC. 

If I hire a employee instead I pay their salary without VAT (so save input VAT) but then can't offset that amount against my output VAT, meaning I have to write a cheque to HMRC that is larger by the same amount as the input VAT I didn't have to pay (and which the consultant would have had to pay to HMRC). 

It ends up a lot like a sales-tax but collected across the supply chain.

The problem with exemptions and zero rating

(I don't want to go into it, but making something VAT-exempt and making it zero-rated are very different things.)

One major problem with zero-rating activities from VAT is that it creates arbitrage opportuntiies if you can shift some or all of the supplies you make into the zero-rated category. 

As most supplies have in some sense a service and a product component, general zero-rating of services would create huge incentives to artificially split out product and service compoments to save on VAT, leading to huge tax bureaucracy and litigation.

That this is a real issue can be seen in the hard-fought court cases over, for example, whether a jaffa cake is a cake or a biscuit or whether shops can save the VAT on a small transactino processing cost component of every transaction.

Ultimately VAT is very similar to a retail sales tax but with collection distributed across the supply chain.