In the attention economy, online platforms are incentivized to design products that maximize user engagement, even when such practices conflict with users' best interests. We conducted a structured content analysis of all Very Large Online Platforms (VLOPs) to identify the designs these influential apps and sites use to capture attention and extend engagement. Specifically, we conducted this analysis posing as a teenager to identify the designs that young people are exposed to. We find that VLOPs use four strategies to extend teens' use: pressuring, enticing, trapping, and lulling them into spending more time online. We report on a hierarchical taxonomy organizing the 63 designs that fall under these categories. Applying this taxonomy to all 17 VLOPs, we identify 583 instances of engagement-prolonging designs, with social media platforms using twice as many as other VLOPs. We present three vignettes illustrating how these designs reinforce one another in practice. We further contribute a graphical dataset of videos illustrating these features in the wild.