Option #1: Oblivion in laudanum.
The solution is extremely Victorian: F.F. Gebrandt's Superior Laudanum, to help you sleep at night. You can find it in the Echo Bazaar (click the "Bazaar" tab), in the shop Nikolas Pawnbrokers. Each bottle costs one Echo, which can be quite pricey if you're trying to sell Whispered Secrets or the like in the Bazaar.
Drinking laudanum is bad for your health, and your Wounds quality will increase a bit one change point), but you can reduce that with F.F. Gebrandt's Tincture of Vigour. I imagine that Gebrandt is an extremely wealthy lady. Alternatively, you can just rest at your Lodgings if your Wounds quality is 3 or higher.
Once you become a Person of Some Importance (PoSI), as with the Tincture of Vigour and Ablution Solution, this option will become less viable; unlike the other two, it will also have a nasty side-affect: every time you drink a bottle of the stuff, you'll become slightly more addicted to it, resulting in the Laudanum becoming less effective.
Option #2: We're all mad here.
Go to A state of some confusion. It will discard and replace all of your Opportunities with new cards, so don't do it if you really like your current Opportunities. Almost all of the Opportunities and actions available in confusion will decrease your Nightmares (though some will abuse your base stats a bit, and one or two will increase your Nightmares while giving you something else you want), and you will leave A state of some confusion once your Nightmares drops to zero.
There are two ways to get to A state of some confusion:
- Go to Mrs. Plenty's Carnival, into the House of Mirrors, and then Gaze Into Dream's Mirror, which requires that you have at least one in Nightmares.
- Let your Nightmares quality increase to 8. You'll be taken there immediately.
A state of some confusion doesn't cost you any Echoes, but has a much worse action-efficiency compared to drinking laudanum. (Laudanum seems to do about 4 change points of Nightmares reduction, and actions in a state of some confusion are more like 1-2 change points of Nightmares reduction.) Not all actions there will decrease your Nightmares, but drinking laudanum will waste some of your time with curing your Wounds, so it's a wash; there's some good reasons to just let yourself go mad. Unfortunately, a few of the actions in confusion have a chance to abuse your four main qualities, and there are a few other small-ish drawbacks, so it's not all upside.
Option #2 if you have Memories of Light: we're all mad here, too.
If you have 10 Memories of Light, then you can spent two of them to go to the Mirror-Marches instead of A state of some confusion at the carnival. If you have at least one Memory of Light, then getting to 8 in Nightmares will take you here instead. The Mirror-Marches are much more generous than A state of some confusion; your main qualities won't be abused nearly as much, and actions that remove two or more points of Nightmares are much more common.
Option #3: Psychotherapy.
In your Lodgings, you'll get an option to Find a way to deal with your Nightmares if your Nightmares quality is 4 or higher. Spend a Sudden Insight (possibly gained via Invite a friend to join you in something terribly intellectual from your Lodgings), and you'll lose 5 change points of Nightmares. Your friend will gain one change point of Nightmares and also spend one action, so you should return the favor sooner or later.
Best Answer
Second chances don't exactly have a cash value; the only way to directly convert them to goods or cash is indeed The End of Battles in the Nadir.
However, that doesn't mean they're useless. There are a number of high-level challenges that are impossible to guarantee success on, even at the level cap. These are listed on the Fallen London Wiki, but a few examples include manufacturing Searing Enigmas and Impossible Theorems at the University (Watchful), "What new madness is this?" on the Listing Tower opportunity card (Dangerous), or the semi-infamous "Where did the Rosers go?" on An Unlikely Garden in the Nadir, which requires 1000(!) Persuasive for a 60% chance of success. The University business, in particular, is extremely punishing -- failing the Impossible Theorem Watchful check will kill you outright, as well as cost you a hefty chunk of change and some Sudden Insights (even if you don't use any).
Second chances were also reworked in late 2015, such that repeatedly failing the same storylet while using Second Chances will very rapidly deplete them (1, then 2, then 4, then 8, and so on) until you either succeed or give up on using Second Chances and fail. As a result, attempting to use Chances in this manner to guarantee success on the high-level challenges will very rapidly deplete your stores.