Such converters exist.
That link goes to the datasheet of the Texas Instruments LM2621. It takes in from 1.2 V to 14V, and puts out a regulated voltage at up to 14V. It can deliver up to 1A.
This is the datasheet of the LT1073. The datasheet includes an example of boosting 1.5V to 9V. It is only intended for low current, though. Like 16mA when boosting 1.5V to 9V. Might be enough if your guitar pedal doesn't need much current.
The things you are looking for exist. They aren't as common as boost converters that start with higher voltage, but they are out there.
Making it work together with an amplifier might be tricky, though. Switching regulators are notorious for "noisy" output. They operate by switching current through an inductor rapidly. The on/off cycles cause "bumps" in the output voltage. These "bumps" can cause audible intereference in audio circuits. The switching frequency of the examples I linked to should be high enough that you can't hear it, but it can still interact with other parts of your gadgets and cause noise.
You can try filtering the output, or you can boost a little higher and use a linear regulator to lower it a bit - that will remove some of the switching noise.
This site gives the power consumption of some common pedals. Some would work with the low current booster, many would not.
Something to keep in mind:
When you boost the voltage, you also multiply the current.
If you need 9V at 100mA and you use a boost converter starting at 1V, then the converter will have to draw 900mA at low voltage. The current goes up by the same factor as the voltage.