You could consider using alternative megastructure/launch system engineering methods to raise your launch loop if you can't build and raise it in place under its own steam.
Given the technology required to build a launch loop, one such bootstrapping system would be the space fountain which uses magnetically accelerated pellets to impart force to the top of a tower via deflector coils, so it isn't entirely unrelated to launch loops. The fountains on their own are handy for space launch purposes, but unlike a launch loop don't provide any speed boost, only a helping hand up in the atmosphere. Fountains have the advantage that extending and contracting them is much simpler as the pellets are connected together into a single giant rotor. Build a series of fountains along the length of your loop to help push the structure into place whilst you're getting it ready, or to provide maintenance later in its operating life.
An alternative would be the colossal magnetic levitation system proposed for the Star Tram gen 2. This is intended to be a launch system in its own right, so you'd need to decide what advantages the launch loop brings over the startram (and there are one or two, and in the limit, launch loops start looking like orbital rings which are all sorts of interesting and useful). Basically you use a whole bunch of giant superconducting cables to generate magnetic fields that repel each other, and some dumb tethers to hold the levitated structure in the right altitude and angle. Astonishingly, you could apparently get the business end the rquired 22km up this way. Scaling to 80km and the weight of a whole launch loop may be rather difficult, but might be enough for assembly and partial raising.
To be honest, I think a Star Tram type system is much easier than a launch loop, and somewhat more straightforward to make and a lot easier to maintain, even if it isn't as cool or powerful. YMMV, of course.
The obvious answer is static scaffolding, but if you could build static structures that high, you could just let that suffice for the final structure and avoid all the trouble and expense of the circulating rotor.
Scaffolding of any kind is going to have some serious limits... even aside from the issue of ultimate material strength and whether you could build a static structure 80km tall, it would require a colossal amount of materials and have a vastly greater environmental impact than a dynamic structure (it would affect weather patterns like a mountain range, and you might need to find an asteroid to provide you with enough metal at a reasonable price and minimal environmental damage). It may actually be easier to build a launch loop, and that's not something you get to say very often.
Whether dynamic (like a fountain) or static, a tower also only gets you up, not out. You still need to achieve orbital speed, and a launch loop provides a means to help you with that in a way that also supports its own weight, killing two birds with one stone.