The Road to Reality  Roger Penrose. Rather ambitiously subtitled 'A Complete Guide to the Laws of the Universe' a concept which would, in itself distress some mathematicians (see next book). I don't know anyone who has read it all but the opening chapters are excellent reading for the Further mathematician with an interest in the subject. 

Goedel, Escher, Bach  Douglas Hofstadter. I loved this book when it came out. I always enjoyed playing Bach on piano and 'cello and I was staggered by Goedel's Theorems. A must for Maths/Philosophy students. 

Proofs from THE BOOK  Aigner and Zeigler. A collection of mathematical proofs which are so neat that they could be found in God's maths book. I don't think that all maths is like this and if you only look at the nice stuff then you miss a lot. Nevertheless this is an excellent read for aesthetically driven maths students. 

A History of Pi  Bekmann  a 1971 classic written for the layman. A must read for any potential student of mathematics. 

e The Story of a Number  Maor another classic. Maor does for e what Bekmann did for Pi. If Pi is about geometry then e is about calculus. These two books sit well together on any bookshelf. (Google books link) 

Five Golden Rules and Five More Golden Rules  both by John L Casti. (link to Cast's personal page). These books survey important developments of twentieth century mathematics and are thus invaluable in understanding what comes next after school in mathematics. The maths study in school could be summed up as mathematical methods and often the most challenging bits date from Euler's time. After doing A level the school student is none the wiser about what has been achieved recently where recently still means the last 100 years. These works put that right. Read them and then tell me you don't want to take the subject further. 
I once had a colleague who told me that he spent his formative years reading the preface only of all the books in his local library. With academic maths books this is a better idea than it sounds. Lots of maths textbooks have a short but stunning synopsis of the subject so far leading up to the content of that particular book. This is often just what you want if you are reading general interest or subject awareness.
The following books are excellent background reading. There are many popular mathematics and science books and you can't go wrong if you start reading any of them which catch your eye. The books I have listed here are 'classics' and I provide relevant links to Wikkipeadia which tells you about the work without trying to sell it to you..