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Is it okay to only fast charge?

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Some say those ill-affects are so minimal with Tesla's battery chemistry that it's not worth worrying about, which is probably why they don't say anything. I think thermal management is the most important thing and Tesla's system mitigates it.
There's also the "public eye" aspect. They don't want to put out anything official that the detractors can use to nitpick and claim degradation is a concern. Instead, they get a big PR benefit saying "just drive the car; charge it often and to whatever level you like; just don't use a blowtorch on the battery" while providing knobs in the car for owners to adjust to their personal comfort level.

Another example is the range/trip charging part of that UI. My recollection is that older firmware would pop confirmation dialogs when you set the then-range-now-trip charge limit (100%) and. The current firmware doesn't do that. I also have a fuzzy recollection that it would either prompt or automatically switch back to Standard (92-93%) charge for the next charging session. Current firmware doesn't do that either.
 
In my "old" Model S I drove 75.000 km during 1,5 years. Have used Superchargers a lot - and have had many 100% charges - including the time to balance the pack. I haven't lost a single KM on the pack since the day it was delivered.

Normally I charge the car to 90% - leave it overnight - and an hour before leaving, using a scheduler, charges to 100%, and pre-heating the pack. My strategy when it comes to charging is: Charge as fast as you use the power. Always leave the car connected. Charge to 80-90% if it should be parked for a period (even overnight). If charging to 100% - never let it sit still. Use is within an hour. Frequently drain the battery almost completely (I am driving it down to 0 km) - and immediately charge to over 20% - or even better do a full charge including balancing.

Can't see that Supercharging should have any negative impact - only positive. Same with CHAdeMO charging.

If you havent seen this (long) video - I would suggest that you do:

Why do Li-ion Batteries die ? and how to improve the situation? - YouTube
 
Frequently drain the battery almost completely (I am driving it down to 0 km) - and immediately charge to over 20% - or even better do a full charge including balancing.

Ack! It's not a good idea to drain the battery to 0 frequently. The damage can be mitigated by not leaving at 0 for long, but it's still better to not drain it further than you have to. I believe Elon said on Twitter that it's best to use the "middle" of the battery. For instance, if you need 50% of the maximum range, change to 75%, and drive it down to 25%.

Draining to 0% then charging to 100% is good for the calibration algorithm (so the car can report the state of charge more accurately), but it's bad for the health of the cells.
 
Ack! It's not a good idea to drain the battery to 0 frequently. The damage can be mitigated by not leaving at 0 for long, but it's still better to not drain it further than you have to. I believe Elon said on Twitter that it's best to use the "middle" of the battery. For instance, if you need 50% of the maximum range, change to 75%, and drive it down to 25%.

Draining to 0% then charging to 100% is good for the calibration algorithm (so the car can report the state of charge more accurately), but it's bad for the health of the cells.
Sitting to 0 is not an issue (given the Model S has "brick" protection). It's the 100% that's the problem.

The other issue is the depth of discharge (difference between highest and lowest SOC). For longer life, it is best to minimize this range. You can think of bending a piece of metal. Bending the metal 360 degrees once (so it is folded in half and then folded in half in the other direction) is more stressful than bending it 180 degrees twice. Similarly, discharging all the way to 0% and charging all the way to 100% is more stressful because there is a larger range of expansion and contraction of the battery material, which degrades it more quickly (cracks are formed more easily).

As for the OP's question, supercharging everyday does degrade the battery faster, but given all the thermal management of the Tesla pack, the effect isn't that noticeable. I don't believe the Plug-in America studies showed any significant correlation between supercharging and degradation (neither did it for the Leaf). If the OP is proposing a CHAdeMO charger (which has much lower power), I think that will be even less of an issue.
 
I would expect it to be fine. You're only talking about ~.6C here - conventional wisdom is that 1C rates are fine for Lithium cells, and even 2C is fairly routine - especially with Tesla's TMS keeping the cells cool.

You'll want to set the charge limit fairly low - not only to avoid sitting with a high state of charge, but also so you can recover all of the energy going down the mountain.