Q:

Xubuntu login loop

I'm running Xubuntu 14.04.3 LTS (trusty).
When I try to login as my main user account I get a black screen for a second, then back to the login screen. I can, however, login as a guest user. I tried separately chowning and deleting ~/.Xauthority as this has been a problem for other people, but this did not help.
If I login using the terminal (Ctrl+Alt+F1) and try to run some commands (i.e. ls, clear, grep), I get the following error for each command:
-bash: /home/cameron/bin/ls: cannot execute binary file: Exec format error

Using /bin/ls or /usr/bin/clear works however, as does using sudo.
This is the output of /bin/cat ~/.xsession-errors:
/usr/sbin/lightdm-session: line 33: /home/cameron/bin/cat: cannot execute binary file: Exec format error
/usr/sbin/lightdm-session: line 34: /home/cameron/bin/truncate: cannot execute binary file: Exec format error
/usr/sbin/lightdm-session: line 106: /home/cameron/bin/ls: cannot execute binary file: Exec format error
/home/cameron/bin/id: 1: /home/cameron/bin/id: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/cat: 1: /home/cameron/bin/cat: Syntax error: word unexpected (expecting ")")
Could not find dbus-launch; Xfce will not work properly

(xfsettingsd:4321): xfsettingsd-ERROR **: Failed to connect to the dbus session bus.

(xfwm4:4322): xfwm4-CRITICAL **: Xfconf could not be initialized

(xfwm4:4322): xfwm4-WARNING **: Missing data from default files
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/cut: 1: /home/cameron/bin/cut: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/cut: 1: /home/cameron/bin/cut: Syntax error: word  unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/cut: 1: /home/cameron/bin/cut: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word  unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/cut: 1: /home/cameron/bin/cut: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/cut: 1: /home/cameron/bin/cut: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/grep: 1: /home/cameron/bin/grep: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/cut: 1: /home/cameron/bin/cut: Syntax error: word unexpected (expecting ")")
/home/cameron/bin/which: 1: /home/cameron/bin/which: Syntax error: word unexpected (expecting ")")
** Message: First Orage start. Searching default timezone.
** Message: Default timezone set to Europe/London.
** Message: Orage **: 15:44:42  wakeup timer init 0
xfdesktop: Fatal IO error 11 (Resource temporarily unavailable) on X server :1.
orage: Fatal IO error 0 (Success) on X server :1.

EDIT:
$ echo "$PATH"
home/cameron/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/home/cameron/.composer/vendor/bin

$ command -v cat
/home/cameron/bin/cat

$ file $(command -v cat)
-bash: /home/cameron/bin/file: cannot execute binary file: Exec format error

$ /bin/file $(command -v cat)
/home/cameron/bin/cat: ELF 32-bit LSB  executable, ARM, EABI5 version 1 (SYSV), dynamically linked (uses shared libs), for GNU/Linux 2.6.32, BuildID[sha1]=b282331acc5bb5f24ebb96445de7695e6d94d0c2, stripped

$ /bin/uname -a
Linux moon-unit 3.13.0-63-generic #103-Ubuntu SMP Fri Aug 14 21:42:59 UTC 2015 x86_64 x86_64 x86_64 GNU/Linux

A:

I think you extracted an ARM root file system in your home directory. You getting all those errors because ~/bin is searched first whenever you try to run something, then instead of run the distribution's binaries in /bin your system is trying to run the foreign binaries in ~/bin.
Try to rename the ~/bin directory:
cd /home/cameron
/bin/mv bin bin.bkp

