If you are talking about classical psychedelics (LSD, Psilocybin, Mescaline, DMT (arguably)) then as far as I am aware there are no reported cases of addiction to date.
It seems that the specific action on the 5HT2A receptors is anti-addictive in nature (Johnson et al., 2014, Bogenschutz et al., 2015). That's not to say that users, (particularly recreational users), don't have or potentially form addictions concurrently with psychedelic use (Carbonaro et al., 2014).
Either as a way of coping with challenging experiences during a trip, or as a result of other pre-existing genetic, environmental or behavioural factors.
Results tend to favour the idea that addiction can be treated with psychedelics (Johnson et al., 2008, Bogenshutz et al., 2015) however managing challenging experiences without proper support can lead to negative mental health outcomes (Carbonaro et al., 2014).
There is an overall favourable view of psychedelics and their role in treating addiction, under the right conditions. With a caveat pointing to the inherent issues that stigma, criminalisation and clinical ignorance play in perpetuating unfavourable outcomes for vulnerable users. By way of cutting off access to resources, either perceived or literally, through the current legal & medical frameworks (Taylor et al., 2016).
Bogenschutz, M. P., Forcehimes, A. A., Pommy, J. A., Wilcox, C. E., Barbosa, P., & Strassman, R. J. (2015). Psilocybin-assisted treatment for alcohol dependence: A proof-of-concept study. Journal of Psychopharmacology (Oxford), 29(3), 289–299. https://doi.org/10.1177/0269881114565144
Carbonaro, T. M., Bradstreet, M. P., Barrett, F. S., MacLean, K. A., Jesse, R., Johnson, M. W., & Griffiths, R. R. (2016). Survey study of challenging experiences after ingesting psilocybin mushrooms: Acute and enduring positive and negative consequences. Journal of Psychopharmacology (Oxford), 30(12), 1268–1278. https://doi.org/10.1177/0269881116662634
Johnson, M., Richards, W., & Griffiths, R. (2008). Human hallucinogen research: guidelines for safety. Journal of Psychopharmacology (Oxford), 22(6), 603–620. https://doi.org/10.1177/0269881108093587
Johnson, M. W., Garcia-Romeu, A., Cosimano, M. P., & Griffiths, R. R. (2014). Pilot study of the 5-HT2AR agonist psilocybin in the treatment of tobacco addiction. Journal of Psychopharmacology (Oxford), 28(11), 983–992. https://doi.org/10.1177/0269881114548296
Taylor, S., Buchanan, J., & Ayres, T. (2016). Prohibition, privilege and the drug apartheid: The failure of drug policy reform to address the underlying fallacies of drug prohibition. Criminology & Criminal Justice, 16(4), 452–469. https://doi.org/10.1177/1748895816633274
Since addiction is a trauma response and use of psychedelics in particular set and settings with good integration afterwards processes trauma, it can be argued that psychedelics are anti addictive. This is not classic recreational use. There's a fair bit of evidence on their use in alcohol addiction for example. There's an issue for some people of spiritual bypassing but they're generally regarded as non-addictive.
there is also caution on the use of psychedelics as many of the reports of the clinical benefits are anecdotal and there are limited clinical trials that have been completed. Over all there is a lack of understanding about the complexity, ethics, licencing, regulatory oversight, public health, and health equity of the medical use of psychedelics.2,3 Administering psychedelics raises understandable safety concerns as these substances have been banned for more than four decades and classified as Schedule I drugs under the Controlled Substance Act (CSA) with no medical use and with the highest potential for use disorder and misuse. Even though these concerns are based on the lack of evidence and the historical association with the counterculture of the 1960s, there are also concerns about not adequately treating mental health conditions and the possibilities that new therapies could bring. (Cut and paste from thesis) I hope it helps Alfred Dearing
Dear Alfred Dearing , I do not know about addiction issues, but I have found one recent research on psychedelic drugs influence on health benefits.
Psychedelic drugs are promising treatments for many mental-health conditions, but researchers don’t fully understand why they have such powerful therapeutic effects. Now, a study in mice suggests that psychedelics all work in the same way: they reset the brain to a youthful state in which it can easily absorb new information and form crucial connections between neurons.
The findings raise the prospect that psychedelic drugs could allow long-term changes in many types of behavioural, learning and sensory system that are disrupted in mental-health conditions. But scientists caution that more research needs to be done to establish how the drugs remodel brain connections...