The HHH Workshop, held in Dubrovnik in July 2023, focused on the progress and challenges in measuring multi-scalar final states, particularly three SM-like scalar bosons at 125 GeV. The workshop emphasized the importance of understanding the Higgs potential and self-couplings to verify or falsify the Standard Model (SM). Key topics included experimental advancements, theoretical studies, and models that can enhance rates compared to SM predictions. The workshop highlighted the need for precise measurements of triple and quartic scalar couplings to determine the full form of the Higgs potential. Despite the low cross-sections for triple-Higgs production, future proton-proton colliders operating at higher energies could significantly enhance these processes, making them experimentally accessible. The workshop also discussed the challenges in $b$-tagging and flavored jets, perturbative calculations, and the use of Monte Carlo tools. The participants agreed to continue discussions and reconvene in Dubrovnik in 2025 to share experimental results on HHH production. The white paper aims to serve as a comprehensive overview of current results and a catalyst for future theoretical and experimental work on HHH production.The HHH Workshop, held in Dubrovnik in July 2023, focused on the progress and challenges in measuring multi-scalar final states, particularly three SM-like scalar bosons at 125 GeV. The workshop emphasized the importance of understanding the Higgs potential and self-couplings to verify or falsify the Standard Model (SM). Key topics included experimental advancements, theoretical studies, and models that can enhance rates compared to SM predictions. The workshop highlighted the need for precise measurements of triple and quartic scalar couplings to determine the full form of the Higgs potential. Despite the low cross-sections for triple-Higgs production, future proton-proton colliders operating at higher energies could significantly enhance these processes, making them experimentally accessible. The workshop also discussed the challenges in $b$-tagging and flavored jets, perturbative calculations, and the use of Monte Carlo tools. The participants agreed to continue discussions and reconvene in Dubrovnik in 2025 to share experimental results on HHH production. The white paper aims to serve as a comprehensive overview of current results and a catalyst for future theoretical and experimental work on HHH production.