I've been working, as a hobbyist, on an alternate method of solar sun shading feel I feel I can help answer this question.
Technically, any project which affect the climate of a planet is a form of geoengineering. This includes the current GHG crisis as mentioned by jamessqf above. But reversing GHG emissions to return to preindustrial levels is also a form of geoengineering. Thus, no matter what we do (continue current trends or modify them) we will be performing geoengineering. The lesson from this is that life is a balance.
What most people think of as geoengineering is a large scale project designed to mitigate certain climate effects across a planetary scale. This includes things like releasing aerosolized particles into the air but also things like increasing seafoam or spraying salt/saltwater into the air over oceans to create small clouds that reflect sunlight.
Something else to consider is that often experiments are conducted specifically to fail. Rocket tests that destroy the rocket have been in the news because they wanted to make sure backups work as expected. Science works that way with geoengineering as well. They try to break things to understand how they break better.
Some things else to consider is the effect on different biomes of different sizes. The world literally operates differently for life of different sizes such as a microbe compared to ants, and an ant compared to us. An ant can fall off the empire state building and be just fine and us, well, not so much. Some insects are so small they swim through the air rather than fly. Thus we have to keep all sizes of life in our consideration. Seafoam may cause certain bacteria to flourish and kill others and we might not know for years. This effect might not be that big or it could cause dangerous algae blooms (this is a theoretical example not backed by any experiment). This plays into your LHC example. LHC operates on the smallest units we can manipulate. The earth is the largest unit we can currently manipulate and the rules are very different.
Going back to your original questions and worst-case scenarios we could cause issues with the earth's ozone layer, collapsing it and wiping out humanity, we might cause massive storms, we might kill off keystone species (some of which might exist on a microscopic level making it hard to detect until it's too late) causing great dyings releasing more GHG into the atmosphere.
The most realistic issue is that we don't apply geoengineering evenlyish (I add ish because optimal applications won't actually be even. Studies have shown that adopting certain areas, or engaging only certain areas with geoengineering, will make negative effects of climate change worse in the least modified areas. This means that is the USA implements geoengineering and Africa doesn't Africa will have a harder time. Conversely, studies have also indicated that well-implimented geoengineering can reduce inequality between nations which would drastically boost the quality of life and economy of the world's population as a whole.
Other than that the next most likely possibility is that geoengineering causes some large scale issues which will cause great dyings which will release more GHG into the atmosphere. Life is a giant cycle of carbon and other compounds and messing with that cycle causes levels of different chemicals and elements to change.
I see the argument that once we start geoengineering we can't stop. This is assuming GHG levels are maintained or increased (which is what we are going for) and suddenly removing a technology that lowers the temp by 0.5-2C would result in fast changes than would happen 'naturally'. However, this concern assumes that we only have 1-2 geoengineering projects to work with. A more realistic view is that we will use multiple methods in conjunction. Not only does this redundancy remove the concern that any single project can't be removed but it allows for some projects being better or safer in different circumstances. We will learn a lot, and it lowers the number of unintended side effects for any particular project.
I hope this helps.