Learn English – Is the word “risk” inherently negative

When in doubt, we can always consult a dictionary. This is what NOAD says (notice how the definition is riddled with nouns like danger, harm, threat and loss, as well as adjectives such as unpleasant, unwelcome and unfortunate):

risk
(noun)
situation involving exposure to danger : flouting the law was too much of a risk | all outdoor activities carry an element of risk.
• [in sing. ] the possibility that something unpleasant or unwelcome will happen : reduce the risk of heart disease | [as adj. ] a high consumption of caffeine was suggested as a risk factor for loss of bone mass.
• [usu. in sing. ] [with adj. ] a person or thing regarded as likely to turn out well or badly, as specified, in a particular context or respect : Western banks regarded Romania as a good risk.
• [with adj. ] a person or thing regarded as a threat to something in need of protection : she's a security risk.
• [with adj. ] a thing regarded as likely to result in a specified danger : gloss paint can burn strongly and pose a fire risk.
• (usu. risks) a possibility of harm or damage against which something is insured.
• the possibility of financial loss : [as adj. ] project finance is essentially an exercise in risk management.

(verb [trans.])
expose (someone or something valued) to danger, harm, or loss : he risked his life to save his dog.
• act or fail to act in such a way as to bring about the possibility of (an unpleasant or unwelcome event) : unless you're dealing with pure alcohol you're risking contamination from benzene.
• incur the chance of unfortunate consequences by engaging in (an action) : he was far too intelligent to risk attempting to deceive her.

I think it's safe to say that, when using the word risk, we are highlighting the possibility of something bad happening, in spite of the possibility for gain.

There is one definition out of those 10 that might leave wiggle room for an exception:

• [usu. in sing. ] [with adj. ] a person or thing regarded as likely to turn out well or badly, as specified, in a particular context or respect : Western banks regarded Romania as a good risk.

but even that exception has a few contingencies:

• it's used as a noun, not a verb
• it requires an adjective
• it requires explicit additional context ("as specified")

Going back to the sentence you asked about:

He risks gaining something more.

I would rephrase that as:

He stands to gain something more.

otherwise, you risk sounding like you're unaware of the word's true meaning.