One might assume, with what she was learning, and the fact that potions would required next to no actual magical power to make, that she might have started with one of the more famous brews, like Draught of the Living Death, or perhaps a bit of Felix Felicis, the former to use on her enemies, the latter to make herself invincible in all things thanks to a surge of good fortune making every opportunity the most optimal.
Hermione Granger, however, wasn't stupid. The former, while requiring only basic ingredients, was still only useful for certain things at the moment. She might need some later, but its shelf life was only a few months, not the years it would take until she might need it. She did have plans involving it, but those plans were subject to change enough that she would put any production of it on hold.
The latter, while one of the nine 'Holy Grails' of potion making, was unsuitable as well. Somewhat because the production of it required esoteric materials she didn't have ready access to. However, more important was that it lacked finesse or control. 'Good Luck' after all, could involve you surviving an accident that kills dozens of people around you, said accident only happening to justify your good luck.
No, her first thought as she set up the still had been healing potions of varying sorts, to be sold to non-magicals. That would provide a source of income, and allow her to test things. Most of the tomes she'd taken into herself had qualifiers that magicals and squibs were the only ones that might benefit from potions in their totality, but that some small effect was still possible for those without magic in their systems, it was merely a matter of degrees and potency.
Better, this would be a test of potion automation, she thought to herself. One of the reasons potions could get so expensive that not everyone had ready access to them was that potions had to be made by hand or wand, you couldn't just set up a system to make them without someone there to monitor. Whether it be the most complicated brew known, or a simple scar removal poultice, you had to have someone doing the work.
Her way, if it worked, would allow her to make potions at industrial level, using a technique she'd read about from an old book describing prohibition in the States. It was a bit more refined than the stills from that era, she hoped. She'd not really studied that subject closely, but she nodded at her work as she set her first batch to brewing, placing some 'Notice-Me-Not' charms on the still, just in case someone came into the bathroom, and assured Myrtle she'd be back the next day.
Said next day saw her up with the dawn, heading to the library, only to stop, and turn around. Her only class, and she had forgotten the book that Professor Quirrel had given her to rush back into her shared room, and quickly pull it out. Hauling it with her, she took it to the library, and made it the first book she 'read' that day, taking in all the knowledge that the Defense Professor had deemed something she should know.
It was…mixed stuff in general. The book was handwritten, as one might expect, but not like the journals she was used to. There was a coldness to the writing. What's more, she was given, without context, some kind of rune formula that she only vaguely recognized, causing her to go through a few books to finally discover it was an array designed to make someone more influenced by suggestions.
That was…odd, but she figured it might have simply been Professor Quirrel giving her a warning about what to look out for, especially given the rest of the book's contents were things that covered the gamut of subjects, but all just this side of Dark. If she didn't know the man as well as she did, she might have even thought he was trying to tempt her to the Dark Arts with it, but given he was not only a Professor at Hogwarts, and thus had Headmaster Dumbledore's approval, but also…well kind of a scatterbrain, she saw it more as just him seeing how far she could go.
That fact that he seemed thrilled when she gave it back to him, not a hint of disappoint that she wasn't speaking in tongues or going on about the rituals she'd taken from the tome said it was just his way of 'preparing' her for the future, giving her a leg up in his class, since she had all the knowledge it could offer same as her other subjects. And she got not only ten more points for her House that day, but an O grade, so she put the book out of her mind.
Mind, she did have some disappointment that day. As it turned out, the water ceased to flow some time after she'd left the bathroom, so her still had failed in its task to make a healing draft. It had, in fact, wasted most of the ingredients she'd added, as they congealed in the night, making her sigh as she vanished the contents, and then added some back in, prepping another batch, and this time, staying in the room to keep the flow of water going.
She had to admit, annoying as that was, she'd at least be on hand the moment this brew finished to judge it. Just as importantly, she had already taken in her books for the day, so she had some things to test out anyway. As such, the moment she confirmed the still was at work, she began to list off, in her mind, all the spells she knew now, categorizing them first by the amount of magic they took to cast.
That wasn't nearly as easy as one might guess at first blush. The amount of effort a spell took to cast was dependent on many, many factors. From the phases of the moon, position of leylines, to simply what color clothing you were wearing. None of them had an individually potent effect on spellcasting, but the dozens and dozens of factors would add up to making it so casting a simple 'Lumos' spell would be like lifting a dumbbell, magically speaking.
That was, she understood, why magic had no formal measuring system. There was simply no way to isolate a single unit of magical energy, and while an individual's capacity could vary, it wasn't more than a few percent between most people. Only a dozen or so in all of history could excel with their power alone, and those were usually through other circumstances. Like a magical embedding jewels in her skin to give her more power, or someone with blood that wasn't quite human allowing them to cast a particular kind of spell with ease.
She wasn't about to try either method herself. The former because that same case had the woman die young, when the jewels, needing a recharge 'fed' off the closest source of magic, that being her body once she could no longer place more jewels into her skin. The latter because blood rituals simply were too unwieldy, and worse, would make other kinds of magic harder to use compared to what they focused on.
Still, annoying as it was to know her first assumption, that wizards had never TRIED to understand magic in scientific terms, was wrong, it gave her a baseline to start from at least. Better, she knew, at least in part, that some spells were harder simply because fewer magicals knew them. The more casters a spell had, the easier it became to cast that spell, like water slowly eroding the ground for a river, which grew larger over time.
That was, as she'd discovered, why so many spells sounded like bad Latin to her. There were other spells, with the same effects, in other regions of the world, that used other words and wand motions. It was simply that repetition shaped the magic, which was…somewhat disturbing, honestly, but she wouldn't dwell on that, as she was a long, long way away from even trying to discover the source of all magic.
After half an hour, which surprised her, she had a list of more than fifty spells she could safely cast. She knew far more than that, of course, the journals, tomes, and books she'd taken in had all had at least one or two in their pages at minimum, usually a dozen or more. Still, she either wasn't sure of specifics in regards to them, in casting structure or effects, or she simply didn't see them as useful compared to the ones she had more experience with.
With that done, she pulled out some papers from her bag and summoned a small table from the floor, grateful that, apparently, none of her tampering with the school set off any kind of alerts or alarms. That book of Rites involving Hogwarts was truly one of her best finds. Still, she set to work mapping out the maths behind each spell she knew, looking for some exploit or another in their inner workings.
She took a break as the sun loomed closer to noon to check on her still, finding this time her procedure had worked, though only to a certain extent. The potion inside had brewed, but it was a weak, soupy mixture, one that would hardly have an effect on a magical or squib, let alone on a muggle. Sighing, she vanished the contents of the still again, and this time began to experiment, adding slightly different ingredients into her concoction, altering it for potency, rather than shelflife.
Leaving to the Great Hall, she listened to the voices of others. Not one was questioning her sudden free time or being jealous or annoyed at her. In fact, no one seemed to notice her at all. She was, if anything, seen as a 'test case' for the Rites, and that someone in the upper years would be the first to REALLY put those to good use, a fact that made her smile, as she realized how few realized just how difficult it would get.
Still, most were opting for private affairs, rather than the public trials she'd gone through. It was seen as the 'proper' way to do things, and if it allowed a student to keep their failures off the lips of everyone else, so much the better. She was already looking around, judging those she thought had a chance of making it, totalling that of those she suspected to be taking the Rites for the various classes, about fifteen-percent would pass.
Interestingly, she had expected most of that to be the Ravenclaws, but rumors stated they were, in fact, the least likely to take the Rites, only two out of the whole House deciding to challenge themselves like that. This was seen as a good thing by some, who said that a Ravenclaw with free time was a Ravenclaw likely to start experimenting, and that was not the sort of thing anyone wanted around the school.
The Slythrens and Gryffindor, probably quite deliberately, were having an equal number in all their classes and years taking the Rites, and looking at who was doing so, Hermione surmised that they would have about equal passing rates as well, given the two Houses, despite what some might say, weren't that far apart in ability or knowledge, though obviously skilled in different areas of each.
Surprisingly, it was Hufflepuffs who looked set to take the most Rites, as they worked hard and studied hard. Where she saw the Ravenclaw duo both failing, due to taking on too much at once, and maybe one each from Slythren and Gryffindor, she saw about half a dozen of the House of the Badger passing, as they were already studying with their groups, and they had better chosen their subjects, doing singular ones, rather than trying to do many at once.
Hermione smiled to herself as she exited the Great Hall, seeing that already, she had made a great change in this school, before her first year was even finished. She had plans for more off course, and as she checked in on the bathroom, finding this time the water had kept flowing while she was out, she added more to the potion mixing, and then got down to the maths behind her spells, to find which ones would be useful for later, and which were merely stepping stones on her road.