We introduce Drivelology, a unique linguistic phenomenon characterised as "nonsense with depth" - utterances that are syntactically coherent yet pragmatically paradoxical, emotionally loaded, or rhetorically subversive. While such expressions may resemble surface-level nonsense, they encode implicit meaning requiring contextual inference, moral reasoning, or emotional interpretation. We find that current large language models (LLMs), despite excelling at many natural language processing (NLP) tasks, consistently fail to grasp the layered semantics of Drivelological text. To investigate this, we construct a benchmark dataset of over 1,200+ meticulously curated and diverse examples across English, Mandarin, Spanish, French, Japanese, and Korean. Each example underwent careful expert review to verify its Drivelological characteristics, involving multiple rounds of discussion and adjudication to address disagreements. Using this dataset, we evaluate a range of LLMs on classification, generation, and reasoning tasks. Our results reveal clear limitations of LLMs: models often confuse Drivelology with shallow nonsense, produce incoherent justifications, or miss implied rhetorical functions altogether. These findings highlight a deep representational gap in LLMs' pragmatic understanding and challenge the assumption that statistical fluency implies cognitive comprehension. We release our dataset and code to facilitate further research in modelling linguistic depth beyond surface-level coherence.