Can one eat crystallized ginger that does not have a hechsher? The ingredients are ginger and sugar, and there almost definetly is nothing else.

All processed food can impose a Halachic challenge of Kashrut, the ingredients are only one aspect, you should check what was on the factories line before, what vessels were used to make the product and other questions. This is why in general food without supervision is out of boundaries. If you think your question is an exception to the rule you should consult with a rabbinic authority that can tell you about your specific item whether it is made in accordance to Halacha or not.