The book I always give as a gift???

A single book? No, not possible Vanessa. You see, it depends on the age and the interests of the recipient - or the purpose of the gift. Well, you know that don't you?
But...babies tend to get Peter Rabbit. All children should own a copy of Peter Rabbit.  It should be the proper little Peter Rabbit with the proper little water colour pictures please.
They should also own AA Milne's poetry "When We Were Very Young" is actually 91 but it is still an excellent introduction to verse and "Now We Are Six" is equally good. Then yes, "The Very Hungry Caterpillar" comes somewhere in between and "The Tiger Who Came To Tea" comes around about then too.
As they head into school and reading for themselves then Paddington is still a favourite for me, along with Pippi Longstocking and the Moomin Books and then, a little later, comic strip type "Asterix" and Tin Tin books.
After that I try Joan Aiken, Eoin Colfer, Roald Dahl and Eva Ibbotson and, for Australian children Colin Thiele and Ivan Southall if they are available. Other people can give them the likes of Paul Jennings.
And for older children still I have offered Neil Gaiman's "The Graveyard Book" - if I think they might be that way inclined. 
I give my godson, who doesn't much care for fiction, books about the human body (preferably funny books), and aircraft. (His mother was not impressed with  him making paper aircraft out of the pew bulletins and launching them in church on Christmas Day - my fault because I gave him the book.)
I give the "how to live on a student budget and still eat" type cook books to teens moving out of home to study. 
I have, with great caution, given Ronald Searle's wonderful "Les Tres Riches Heures de Mrs Mole" to people I have felt needed it.
I have given Dag Hammarskjold's "Markings" to others. 
And, for many children, I have at some point given them a large, hard covered completely blank book and some pencils and told them "This is your book to write or draw or both exactly as you wish."
Everyone should have one of those.