There seems to be quit a few pure blood families in the books. Is there any list or reference for how many there are? I'm asking because there is no mention of the family Potter other than James and Harry. Harry seems to be at least a quarter muggle if not more. Since there are so many wizards concerned with keeping the wizard lines pure (an extreme example is the Slytherin line) it seems there would be some record.
0) Side note: an interesting fact noted by someone on the Wiki: By the conclusion of the series, the Weasley family is the only known pure-blood family to have several male heirs :)
1) There seem to be a total of at least 34 pure-blood (Mostly British) families mentioned. This should start you off: http://harrypotter.wikia.com/wiki/Category:Pure-blood_families, with this helping: http://harrypotter.wikia.com/wiki/Pure-blood#Known_pure-bloods (the second one has more).
Here's an incomplete list:
Blishwick Brown Bulstrode (Millicent was in Slytherin) Burke Cornfoot Crabbe Crouch Flint Gamp Gaunt (or should they be counted as Slytherins?) Goyle Greengrass House of Black Lestrange Longbottom (Neville) MacDougal Macmillan Malfoy Nott Peverell Potter Prewett Prince Rosier Selwyn Slytherin Weasley Yaxley
2) Please note that the families are all inter-related, e.g, the Black family tree shows that the following families from the above list are ALL related:
Blacks Bulstrode (NOT on Wikia list) Burke Crabbe Crouch Gamp Lestranges (married Bellatrix) Longbottom (NOT on Wikia list) Malfoys (Married Narcissa) Macmillan Potter Prewett Rosier Weasley Yaxley
3) As an aside, a couple of corrections to some statements in your question:
there is no mention of the family Potter other than James and Harry.
No. There was a Charlus Potter married to Dorea Black/Bolstrude on Black family tree. Also, JKR mentioned Harry's grandparents in her interviews, they largely financed anti-Death-Eater activities, and died of old age and ordinary wizarding ilnesses before Harry's parents died (on purpose, since JKR needed Harry to be alone in the world).
Harry seems to be at least a quarter muggle if not more.
Since Lily was referred to as "Mudblood", she was a full on Muggle. So Harry is at least 1/2 muggle, not 1/4 (if you don't count weird genetic drifts that implied some wizarding blood early on on Evans Muggle family - as per JKR, all Muggle born wizards have a wizard ancestor)
A late answer, but new information on the British pure-blood wizarding families has come out on Pottermore from J.K. Rowling herself.
In the early 1930s, a 'Pure-Blood Directory' was published anonymously in Britain, which listed the twenty-eight truly pure-blood families, as judged by the unknown authority who had written the book², with 'the aim of helping such families maintain the purity of their bloodlines'. The so-called 'Sacred Twenty-Eight' comprised the families of:
¹I personally do not recall either the names "Fawley" or "Shafiq" appearing in the books -- please correct me if I'm wrong. I'm not going to go over any of the HP Wikia information; the above list of British pure-blood families is fairly definitive to my inner canon nerd.
²Widely believed to have been written by one Cantankerous Nott