Using outdated versions of Adobe Experience Manager (AEM) carries general risks similar to any software that is not regularly updated. It is essential to keep your system secure, stable, and performing optimally by staying up to date with the latest AEM versions. Additionally, updating allows you to take advantage of the latest features and support resources provided by Adobe.
IMHO the major risk is that you will be on your own, and nobody will give advice/support, because issues may be fixed on newer versions.
My recommendation would be to first, go through the list of new features each version AEM brought in 6.4 and 6.5 and see if something of that is critical to your use case. Second, check which are the upcoming features in your roadmap until the time the upgrade happens and understand if there is something that can be critical to your AEM structure, for example, 3rd system integration, etc. Then of course plan as soon as possible the upgrade, given that you are on 6.3, the transition to 6.5 should be smooth (unless you have a lot of customization/overlays).
Adobe has provided these tools for migrating and upgrading faster: https://experienceleague.adobe.com/docs/experience-manager-65/deploying/upgrading/pattern-detector.html?lang=en
