Here is one famous quote from Steve Jobs :
“Most people make the mistake of thinking design is what it looks like. People think it’s this veneer – that the designers are handed this box and told, ‘Make it look good!’ That’s not what we think design is. It’s not just what it looks like and feels like. Design is how it works.”
Personally, I don't think that there is any website without a design, but there might be websites which are not styled with glittery gradients and fancy illustrations. All websites are designed to to do some or another task but the question is, are the designs solving the problem that it is intended to do. This along with the value, the website/product offers to its users determines the success of the website/product.