I love the concept of experimenting, but it appears to be so hazardous I have been nervous about using it. Number of botch dice equal to the magic aura? Plus one for each point that you're pushing yourself?
If you live in a level 7 aura and do a +3 push, that's 10 botch dice! Yikes! And some of the bad outcomes are REALLY bad. Heck, some of the "good" outcomes are pretty bad.
How often do people use experimentation, and under what circumstances?
Unless you have Vis to burn, I generally recommend experimenting only when inventing spells. Altho' a failure with any Lab Activity will flush the time spent down the drain, at least no vis is lost.
However, there are 3 reasons to experiment:
If it's #1, then Spells serve as well as any other, and risk less.
If #2, you may not have any other choice.
If #3, you shouldn't let considerations like Vis slow you down.
And always wear your safety goggles.
(Note- Even a lot of the "failures" aren't actually all that bad, at least if you aren't investing (much?) Vis in the effort (ie, for Spell invention only). While reduced/modified/limited effects are disappointing, a lesser spell can still be useful to the creative mind. And it often still acts as a Lab Text to add to that Lab Total for trying again next time (with or without experimentation.)
Otoh, experimenting with a major invested magical item, anything over several seasons, an existing pre-invested talisman, investigating items, or (ulp!) when enchanting your familiar is just askin' to get beaten down. Murphy's Law lives for such moments.)
Cautious Sorceror is your friend.. as is the golden cord on your familiar, and the safety rating of your lab 
With enough effort, you can get that 10 down to a much more reasonable number ^^
I've always had a weak spot for experimenting. Of course, I'm usually the storyguide, so I have characters to burn 
Imagine a lab accident during experiments with a target:bloodline spell, for example. Mwahaha.
I'm afraid my first magus as a player will die an early death from his/her nocturnal addled triple-overtime experiments in his gremlin-filled disorganized mountaintop eyrie.
Indeed. Although I don't beleve you can lower it below 1.
Experimentation is especially useful for Original Research, indeed required. Otherwise, I would agree it's best to experiment while inventing spells to save on raw vis and time, especially on spells that you can invent in a single season (with experimentation). Lesser enchanted devices are also viable, not swallowing up too much time and not ruining the existing item, if raw vis is plentiful.
Some little calculations:
If you have 3 botch dice and no risk modifier:
Chance on failure (complete failure/flaw/modified effect/botch): 25,3 %
Chance on bonus (positive side effect/discovery): 7,7 %
Disaster chance: 3,4 %
If you have 3 botch dice and risk modifier +1 (so actually 4 botch dice):
Chance on failure: 27,5 %
Chance on bonus: 15,4 %
Disaster chance: 5,9%
The chance that your season & vis is wasted or that you're with a bad or useless spell or effect is pretty high. But if you do experiment, use 1 risk modifier, the chance that you gain a discovery is increased very much and the 15 xp are really nice.
15 xp not worth the risk.
But isnt the potential story worth it?
We have a magus with charmed life who tends to experiment all the time. He has been lucky so far and only had one or two mishaps. A side effect here or there, but thats about it.
My magus only experiments if he really needs the lab total (generally to make an item in a single season, to save vis). His apprentice has inventive genius so experimenting in the labb can produce MUCh higher lab totals.
+6 for inventive genius
+3 for full risk modifier
thats an average of 14 on my lab total. Nothing to sneeze at.
Admittedly, on a botch we are looking at 8 botch dice. Which could really hurt, but i tend to trust to the god of lucky dice.
Practically, the inventive genius only gives a +3 because the other 3 you also get without the experimentation. And with the higher risk modifier the chances of extraordinary results are increased too. So, I do not know if this is really handy.
Get a lab with a good safety rating - which isn't at all hard to do... Just take the time to improve it a little, refinement alone gives a very healthy safety for a bonisagus...
My magus has had only good experiences with experimention, even using it on his familiar binding (need a high total to make a bound at all...)
My charcters have never had good luck experimenting. Their creations always end up with undesirable side effects and flaws.
I think that the time when experimenting is most useful is when it makes the difference between getting a project done in one season and gettign the project done in two seasons. If the experiment works you save a season and may even learn something, if the experiment doesn't work to your satisfaction more often then not the character has at least some version of the spell that can be used for a similar spells bonus that will allow you finish the spell in the second season anyway.
Yeah, and if the first spell isn't entirely corrupted, it can have uses that the intended one didn't quite aim for. Sometimes, if you're creative, and lucky. But first lucky. 
And I hope whoever is running your familiar never lets you live it down!!! 
Well, my little dragon was just a few years out of her shell at the time, and the bond ended up with a side effect allowing us constant mental comunication... (saving me 2 seasons of enchanting) - And I'm running her myself...