At the risk of sounding like a parrot, LOTR and Narnia both influenced me as a child. Later, I got to add to this collection The Name of the Rose by Umberto Ecco and The Life of Pi by Yann Martel.
The Name of the Rose is a book about a 13th-century monastery in northern Italy which is subjected to a series of horrific events, including murder. Around the story, though, are terrific debates between monks of various orders as to the nature of God, the acceptability of mirth, and other complicated issues. It's a great mystery story, too, as the protagonist tries to put together these seemingly connected, seemingly disparate events.
The Life of Pi is a true (maybe) story about an Indian boy, Pi Patel, whose family ownes a zoo. They begin to lose money and decide to move to Canada. On the way, the Japanese barge they hired sinks, and only Pi and a tiger they planned to sell in Canada survived. There, for months on end, Pi is stranded with the tiger on a small lifeboat, and has to avoid getting eaten among various perils of the South Pacific. In the end, he recounts the story to the Japanese bureaucrats who are sent to him to discover what happened, who are baffled by his strange tale. Pi tells them another, more believable story, but that one actually sounds even more unlikely. Pi asks them, since they cannot tell which is the true story, if either, which one they liked better. They responded, "The one with the tiger." Martel, in his prologue, says that he met Pi in Toronto a few years ago, and he wrote the story exactly as Pi had told it. These bureaucrats seem to echo Martel's and the reader's questions about the veracity of this tale. Nevertheless, it is exciting, philosophical, and very well-written.