It is often said that you cannot consent to being murdered. If you say to someone 'Please kill me' and they do, they can still be tried and convicted for some degree of murder or manslaughter. Furthermore, in many countries, assisting in suicide is also considered a crime, even if the person is fully mentally competent.
Are there any other crimes that the victim cannot consent to, in order to prevent them being a crime? It's worth adding that the consent is freely given with no coercion.
A problem with the question is that it uses the loaded term "victim". If you change the question to "Are there actions that you can perform involving another person, which are crimes even if the other person consents to participating in the action", then there are very many. Selling alcohol to a minor; selling heroin to anyone; selling sex in most US jurisdictions; selling firearms to a convicted felon. Also, for a physician to assist a person in suicide, in most states. Formerly in the US, various forms of sexual intercourse were acts that consent did not make legal. Whether or not the person is a "victim" in these cases is open to debate.
In the case of physician-assisted suicide in Washington, the doctor is allowed to prescribe (oral) drugs that the person ingests: only a doctor is permitted to do this, both under the DWD Act and as a consequence that only a doctor can prescribe the drugs. There seems to be a belief that it is a crime to assist a person in committing suicide, which is probably correct if the assistance is shooting the person in the head, or in general directly causing the death (thus, "I give you permission to shoot me in the head" doesn't cut it). But from what I can tell, it is not generally against the law in Washington to help a person who commits suicide (e.g. supplying the means of self-dispatching). In some countries, suicide and helping with suicide is illegal, e.g. in Kenya Penal Code 225 says
Any person who (a) procures another to kill himself; or (b) counsels another to kill himself and thereby induces him to do so; or (c) aids another in killing himself, is guilty of a felony and is liable to imprisonment for life.
No exception is created if the person consents to being aided to kill himself.
A victim can consent to no crimes. The decision of what is and is not a crime is solely the prerogative of the state.
Now, there are some actions that may be criminal if done without permission and not criminal if done with permission - that is not because you have permission to commit a crime; it's because the action is only criminal in the absence of permission. For example, taking someone's property is theft in the absence of permission, it is receiving a gift with permission. Hitting someone is battery without permission, it's a contact sport with permission.
The last example is particularly illustrative. A contact sport has rules (explicit or implicit) about what form of contact is allowed, if you recklessly or intentionally cause harm outside those rules then you have stepped outside the limits of the "not a crime" in into the "crime".
Related: Does a police officer have to get permission to break the law?
