Hmm....ok. What about someplace else while I'm there, like France or Spain for example?
You would have to look into the immigration requirements for marrying in another country, as you may also need a visa to marry in either France or Spain (or wherever else you're thinking of).
Well after much more goggling, it seems it would be much easier and less likely to get in trouble with immigration, to just come back on a fiance visa. So much for running off and eloping 
If she is French, then you don't have to go the fiance visa route if you don't want. As she has been in the UK for 17 years, she is a permanent resident in the UK (does she have UK citizenship as well, or just French?), so you can either go the EEA visa route or the UK visa route.
UK visa Route (fiance visa)- The fiance visa is valid for 6 months and you have to marry and apply for your next visa within the 6 months. You cannot work at all while on the fiance visa and it costs £810 ($1,337).
- After you marry, you need to apply for a 2-year extension visa to live in the UK - this allows you to work and costs either £550 by post (up to 4 months processing before you can work) or £850 in person (same-day decision).
- After you have been in the UK for 2 years on the FLR(M) you will apply for ILR (permanent residence) - this currently costs £972 by post or £1350 in person.
Total cost of visas to ILR (current prices) is between £2332 (postal applications) and £3010 (in person applications).
All of these fees increase twice a year, so by the time you are eligible to apply for each one, they may be much more expensive.
UK visa route (marriage visitor visa)- You can get a marriage visitor visa which allows you to marry in the UK, but then you would have to return to the US to apply for a spousal visa to move to the UK, because you cannot switch to another visa in the UK as a visitor. The marriage visitor visa costs £76 ($125).
- Once you are married, you return to the US and apply for a 2-year spousal visa, which allows you to work in the UK. It costs £810 (same as fiance visa)
- After 2 years in the UK on the spousal visa you can apply for ILR (currently £972 or £1350)
Total cost of visas to ILR is between £1858 (ILR by post) and £2236 (in person ILR)
EEA visa route- To marry in the UK you can get a marriage visitor visa (£76). On the EEA route, you don't have to leave the UK to get the next visa and you can just apply to switch in-country.
- After you are married, you can stay in the UK and apply for a 5-year EEA residence card. This is free.
- After living in the UK for 5 years, you can apply for permanent residence. This is also free.
Total visa cost to permanent residence = £76 (for marriage visitor visa) or whatever it costs to marry in another country.
If you want to marry in another country you can (or if you want to return to the US after marrying in the UK on a marriage visitor visa you can), but you would need to return to the US to appy for an EEA family permit in order to enter the UK. The EEA family permit is free. Once you have moved to the UK, you then apply for the 5-year residence card (free) and then permanent residence after 5 years (free).
The only drawback to the EEA route is that it takes 5 years to permanent residence, not 2 years (UK visa route). However, people with EEA residence cards have more rights and benefits than people on UK fiance or spousal visas, so it's not such a big issue and it will save you thousands of pounds on visa fees compared to the UK route!
