What’s the simplest instruction set for a compiler that can compile itself? I think you probably need some kind of looping construct, some kind of way to read in bytes, some kind of way to output bytes, and probably sequencing of statements. You can probably avoid much parsing by making the input instructions be single bytes, but you probably still need some kind of arithmetic to calculate the jump offsets.
Mats Linander’s Awib is a multi-target optimizing BF compiler written in BF, with six backends (it can output Linux i386 ELF executables, C, Tcl, Golang, Ruby, and Java), so BF is clearly a sufficiently powerful language to write a self-compiling compiler in. (It also contains an ASCII-art portrait of Meriday in the morning and can be compiled as Tcl, bash, or C, as well as BF. Truly impressive.) It’s about 43 kilobytes, and it would presumably run under Urban Müller’s original 240-byte AmigaOS BF compiler or Brian Raiter’s 199-byte Linux version: http://www.muppetlabs.com/~breadbox/software/tiny/bf.asm.txt
Daniel B. Cristofani has written a much more minimal BF self-compiler, targeting C; I found a copy at http://esoteric.sange.fi/brainfuck/impl/compilers/dbf2c.b, and he has a copy on his own web site at http://www.hevanet.com/cristofd/brainfuck/dbf2c.b. It seems to work; anyway, I compiled it with itself, verified that it produced the same output when compiled with the self-compiled version of itself, compiled Linus Åkesson’s Game of Life with it, and played Life successfully on the result. Cristofani’s self-compiler for BF is 1183 bytes, but running Erik Bosman’s bfstrip utility from http://esoteric.sange.fi/brainfuck/utils/bf-tools/bfstrip.c on it reduces it to 904 bytes.
So, one approach to bootstrapping things from BF would be to compile programs from other languages into BF, and then run them with one of these BF interpreters. But I think there’s a much more interesting approach available, which is to add some instructions to a self-compiling BF implementation, recompile it with itself, and then update it to use the new instructions.