The phrase "conceivable kind" refers to any possible category or type of something that can be imagined or thought of. It means considering all the different options that can be imagined or conceived. Full definition
See cute photos of children and their blankies, meows, robots, and every other conceivable kind of comfort object. (babycenter.com)
The term is used to refer to every conceivable kind of problematic content, «including both blatantly and accidentally inaccurate information, salacious and fear - mongering headlines, hateful and incendiary rhetoric produced in blogs, and propaganda of all stripes.» (fortune.com)
If we rejoice in rendering such practical service abroad, why should not our churches at home be centers of every conceivable kind of helpfulness in their communities? (religion-online.org)