The answer depends on how literally you intend the wording. Technically, you can't make "soap" from scratch without lye (it's how "soap" is defined; it isn't just broken-down fat, it is a specific type of chemical salt resulting from a certain reaction). However, you can make materials from scratch that behave like soap without using lye, or you can start with a soap base where someone else has already done the "dangerous" part.
Let me pick apart the assumptions behind the question a little to get to information that answers what you want to know.
How do you define "soap"?
I'm guessing from this and your other recent question on soap, that your objective is a DIY material that has the cleaning properties associated with soap (perhaps with the additional requirement that it is solid at room temperature). Soap is one of the materials like that, and it is popular to make from scratch. But it isn't the only one and there is more than one way to get to the end result.
Most commercial bars of soap aren't actually soap (if they are called something like "bath bar", "beauty bar", a familiar product brand name, or something else, and don't include the word "soap", they aren't soap). They are likely synthetic.
Perspective on the dangers
There are several elements to the danger posed by lye. One is in the handling and the soap-making process. Lye is caustic. It can cause chemical burns if you get it on your skin and it can ruin your clothing or the finish of surrounding surfaces if you splash it on them. So you need to take reasonable precautions, be careful, and pay attention to what you're doing when you use it.
The other danger is unreacted lye in the finished soap, either because you started with an inaccurate mix or the reaction hasn't yet gone to completion. This will be much more diluted than the concentrated lye used to make the soap, but it can irritate your skin or worse. So you need to use an accurate mix and give the soap time for the reaction to complete. You can also err on the side of using a little less lye than required, which will leave a little fat not converted to soap.
However, keep some perspective on the danger. You may already use concentrated lye if you use drain cleaner. If you eat pretzels, they have received a wash of diluted lye. If you get lye on your skin, you can wash it off and/or neutralize it before it burns your skin off, so you might just get a little irritation if you're fast. People have been making soap with lye for thousands of years without disfiguring themselves. Bottom line: it's reasonably safe if you treat it with respect. Still, if you don't want to mess with it, there are alternatives.
Can you make soap from scratch without lye?
Soap is defined as a product made from animal or vegetable fats reacted with lye. It takes a strong alkali to produce the chemical reaction (the reaction won't happen with a weak alkali, so you can't use something like baking soda and just give it more time); it requires both a very active alkali and a high concentration of it.
Soap is also a specific chemical salt. There are other strong alkali materials that might break down the fat, but the end product wouldn't necessarily behave like soap (and you would still be using a strong alkali with the dangers you want to avoid). Basically, if you want to make soap from scratch, the reaction requires a strong alkali, so any suitable alkali would carry the same risk. A safer alternative won't work.
The main DIY soap options
Make primitive soap from scratch. This goes to the ashes mentioned in the question. You can make crude soap the way it was made for thousands of years until recent times. Use animal fats or vegetable oils and extract the lye from ash. People used a few tricks to get the lye concentration in the right ballpark (like floating an egg or potato in it to see how much it sank). But the process is crude and very variable. It's easy to get soap that is irritating to use, or results having off odors or color.
This isn't a no-lye process, although lye extracted from ash isn't likely to burn your skin off immediately on contact. The results can turn out pretty nasty, though; you aren't likely to be giving bars of this stuff as gifts.
Make modern soap from scratch. The main difference is precision. The lye is pure and a known concentration. The oil is pure. You can consult reference materials for the exact ratio of lye to oil based on the specific oil. The results are very predictable and consistent, and the soap isn't harsh. However, this is the option you are looking to avoid.
Make modern soap starting with a soap base. With this option, someone else has already done the dangerous part and produced the basic soap component. You modify that to produce the specific characteristics you want and mold it into bars or a decorative form. This is a pretty popular approach.
Make detergent from scratch. "Detergent" doesn't refer here to a product used in a washing machine or dishwasher. I'm using it generically to refer to a soap-like product made directly from synthesized or refined chemicals that perform the functions of soap and have the characteristics you want. The term is sometimes used more restrictively for a specific class of materials (potassium or sodium salts of a long alkyl chain ending with a sulfonate group; but don't get hung up on that). Detergents can be better than soap in that they don't bond with the minerals in hard water to form a scum like soap can. This option may appeal more to chemists than crafters.
Use plants or make plant-based "soap". There are a number of plants that contain high amounts of a material called saponin, which behaves like soap. In some cases (like soap nuts), you can directly use the plant like soap. In other cases, you can easily extract the saponin and use it like liquid soap. 9 Natural soap plants for making lye-free soap is a good place to start for this option.
So if you want to avoid working with lye, there are still DIY "soap-making" options.