The Sale of Goods Act 1979 gives you certain statutory rights when you buy something that the retailer, in this case Argos, are responsible for fulfilling. This includes what they must do if you receive faulty goods and is separate from any warranty you may have.
More info here:
http://www.moneysavingexpert.com/shopping/consumer-rights-refunds-exchangeThe key for you is that any goods must be fit for purpose and last a satisfactory amount of time. It sounds like the laptop doesn't meet these criteria.
I'd write to Argos Customer Service first, explaining that the laptop isn't fit for purpose despite being in for repair a number of times and requesting either a full refund or a replacement as the repairs have obviously been unsuccessful.