Niccolò Paganini

Wikipedia (1782-1840) [Italian. Classical.]
If you're just gonna listen to one: Sonata for Violin & Guitar (op. 2 no. 1)

Background

There's a lot of funky energy around Paganini - his story is the stuff of legends. Some accounts say that Paganini mutilated his hand in his youth, to give him a better reach when fingering on the violin. Other accounts of his life suggest that he might have had Marfan Syndrome, which can manifest in too-limber joints and very long limbs, and sometimes spider-like fingers.

However he came to them, he had fingers uniquely suited to playing the violin, and he was moved from teacher to teacher as a boy, searching for someone who could actually teach him something. He was a prodigy, and his technical abilities with stringed instruments eventually made him famous.

There was no music complicated enough to showcase his skills, though, so he composed his own music. His music is not thought of as highly interesting, but exists primarily to show of technical skills of the violinist.

Orchestral Music

Concerto No. 1 (Uto Ughi)
Concerto No. 2 (Ivry Gitlis)
Concerto No. 4 (Salvatore Accardo)

Caprices

Caprices 1, 5 & 24 (Itzhak Perlman)
Caprices 1-24 (Alexander Markov)

[The theme from Caprice 24 has been used by many composers to create other works, most notably: Variations on a Theme of Paganini (Brahms), Grandes études de Paganini (Liszt) & Rhapsody on a Theme of Paganini (Rachmaninoff).]

Chamber Music

Paganini wrote lots of chamber music, in many different instrument combinations, all of which could highlight his virtuosic capabilities (as a violin, viola, and guitar player -- he excelled at all of these instruments). Some examples:

Nel cuor più non mi sento (usually violin with violin & cello accompanying)
Cantabile (violin & guitar)