Build your own web browser from scratch. Implement HTML parsing, CSS styling, layout computation, and rendering. This is one of the most ambitious projects you can tackle, touching nearly every area of computer science.
A web browser is an incredibly complex piece of software that combines networking, parsing, layout algorithms, graphics rendering, and scripting. Building even a simplified version gives you a deep appreciation for how the web platform works and how browsers turn raw HTML, CSS, and JavaScript into interactive pages.