Patient gets to keep their canine then.

That’s a good point. I remember seeing an article about tooth re-growing teeth (in ferrets), and while I don’t remember if it was stem cells, that might be nicer than having to lose a tooth for an eye.

Do you know if tissue grown from a patient’s own stem cells is generally not rejected by the immune system

My background is a bit limited here, but looking around it seems that it’s ‘better’ but not necessarily ‘rejection proof’

HSCT came to mind first, but those are replicated inside the patient:

Hematopoietic stem-cell transplantation (HSCT) is the transplantation of multipotent hematopoietic stem cells […] in order to replicate inside a patient and produce additional normal blood cells. HSCT may be autologous (the patient’s own stem cells are used), syngeneic (stem cells from an identical twin), or allogeneic (stem cells from a donor)

Autologous transplants have the advantage of lower risk of infection during the immune-compromised portion of the treatment, since the recovery of immune function is rapid. Also, the incidence of patients experiencing rejection is very rare (and graft-versus-host disease impossible) due to the donor and recipient being the same individual

Induced pluripotent stem cells seem closer:

Since iPSCs can be derived directly from adult tissues, they not only bypass the need for embryos, but can be made in a patient-matched manner, which means that each individual could have their own pluripotent stem cell line. These unlimited supplies of autologous cells could be used to generate transplants without the risk of immune rejection. While the iPSC technology has not yet advanced to a stage where therapeutic transplants have been deemed safe, iPSCs are readily being used in personalized drug discovery efforts and understanding the patient-specific basis of disease.

This other article from 2013 lists a few concerns, and I think this is the closest to what you were looking for: https://pmc.ncbi.nlm.nih.gov/articles/PMC3931018/#sec3

Potential Causes of iPSC Immunogenicity

[…] The first potential cause is immaturity of cells differentiated from iPSCs in vitro. […] There are a number of human cell types that, to date, can be differentiated only to immature phenotypes in vitro […] . An immature phenotype poses two risks for immune response, the first being low MHC class I (MHC-I) expression. Natural killer (NK) cells target cells with low MHC-I levels, and although differentiation of iPSCs causes these levels to rise, they may not reach those of adult tissue. […] Another risk of an immature phenotype is expression of embryonic or fetal proteins. These antigens may not have been present during immune system education to go through negative selection in the thymus, leaving them susceptible to T cell attack. T […].

A second potential cause of iPSC immunogenicity is genetic and epigenetic changes that arise from reprogramming or adaptation to culture conditions. Recent studies have demonstrated that reprogramming to pluripotency is incomplete and that iPSCs carry an epigenetic memory of their tissue of origin that affects gene expression and can restrict differentiation potential (26–30). […]

A third potential cause is culturing of iPSCs, or their differentiated progeny, with xenogeneic or non-physiological culture reagents. […] hESCs take up the non-human sialic acid N-glycolylneuraminic acid (Neu5Gc) from mouse cell feeder layers and animal serum-containing culture media. This represents a risk because humans have circulating antibodies to Neu5Gc (37). Several groups have since developed xeno-free culture conditions for reprogramming and differentiation that reduce or eliminate Neu5Gc expression, although these methods are costly and can be technically challenging (38–40). […]

[…]

A fun fact I came across on that wikipedia article:

Yamanaka named iPSCs with a lower case “i” due to the popularity of the iPod and other products.