"The Mortal Immortal" is narrated by Winzy, who tells the story of how, over 300 years ago, he worked as an assistant to the alchemist Cornelius Agrippa. Frustrated by his lover Bertha's demands to marry even though he was too poor, he drank a potion that Agrippa was preparing, believing it to be a cure for love. It had the opposite effect, however, increasing his love for Bertha and giving him the confidence to marry her in spite of his uncertain prospects. Five years later, Agrippa called him to his deathbed and asked for his help preparing another batch of the potion, and revealed that it what Winzy drank was actually the Elixir of Immortality. Winzy is not sure whether he is really immortal, but as the years pass he still looks and feels like a young man while Bertha ages. After Bertha's death, and as the story ends, Winzy determines to travel to the far corners of the earth to test his immortality in Earth's harshest climates.
This the most widely read and most commonly anthologized of Shelley's short stories, perhaps because its exploration of the idea of immortality and the ending in which Winzy is travelling to the ends of the earth is reminiscent of Frankenstein.
For more information about this tale, visit its Wikipedia page.