Large Language Models (LLMs) can be driven into over-generation, emitting thousands of tokens before producing an end-of-sequence (EOS) token. This degrades answer quality, inflates latency and cost, and can be weaponized as a denial-of-service (DoS) attack. Recent work has begun to study DoS-style prompt attacks, but typically focuses on a single attack algorithm or assumes white-box access, without an attack-side benchmark that compares prompt-based attackers in a black-box, query-only regime with a known tokenizer. We introduce such a benchmark and study two prompt-only attackers. The first is an Evolutionary Over-Generation Prompt Search (EOGen) that searches the token space for prefixes that suppress EOS and induce long continuations. The second is a goal-conditioned reinforcement learning attacker (RL-GOAL) that trains a network to generate prefixes conditioned on a target length. To characterize behavior, we introduce Over-Generation Factor (OGF): the ratio of produced tokens to a model's context window, along with stall and latency summaries. EOGen discovers short-prefix attacks that raise Phi-3 to OGF = 1.39 +/- 1.14 (Success@>=2: 25.2%); RL-GOAL nearly doubles severity to OGF = 2.70 +/- 1.43 (Success@>=2: 64.3%) and drives budget-hit non-termination in 46% of trials.