First example that comes to mind is George R.R. Martin's 'A Song of Ice and Fire', specifically chapters from Eddard Stark's perspective. We're locked inside his head, hearing his thoughts and judgments, but we only see what he sees and know what he knows. The world is filtered through his honor-bound, Northern lord sensibilities. We feel his growing dread in King's Landing, his misinterpretations of people like Littlefinger, but we're never given an omniscient narrator to correct him. That's the core of it right there – the limitation creates dramatic irony and tension. The reader pieces together the larger conspiracy from Ned's fragmented, biased view, which makes the eventual payoff so much more impactful than if we'd been following Cersei or Varys around getting the full picture.
Another fantastic, more intimate use is in Kazuo Ishiguro's 'The Remains of the Day'. The entire narrative is Stevens the butler's recollections, and the limited perspective is the entire point. We only get his highly repressed, professionally dignified interpretation of events. His feelings for Miss Kenton, his father's death, Lord Darlington's politics – all are reported with a stiff upper lip. The reader has to actively read between his lines, decoding the immense emotional turmoil he refuses to acknowledge. The power isn't in what Ishiguro shows, but in what he forces the reader to infer from what this specific, limited consciousness chooses to report and how he phrases it.