Yes - on my network, Vodafone UK, these are the ones I can remember off the top of my head:
*#1345# - gives you your credit balance if you're on a prepaid account
*#100# - your phone number
*#101# - the current network date and time
*#[102-105]# - various network engineering information that I can't understand.
You can also, as long as you're on original GSM and not 3G, use the 'cell broadcast' feature to make it show you the current area code you're in (or more accurately, that your current tower is in). This is a throwback to an ancient price plan which gave cheaper calls to your local area. It's very much deprecated, so some newer cells don't broadcast the area code info, and no 3G cells do so.
On Android it often depends on the specific model you have. Here are some for the Samsung S3:
#06# Show IMEI number
#0# LCD Test Menu
##4636##* user statistics and Phone Info
#0011# Displays status information for the GSM
#1234# View SW Version PDA, CSC, MODEM
#12580369# SW & HW Info
#197328640# Service Mode
#32489# (Ciphering Info)
#232337# Bluetooth Address
#232331# Bluetooth Test Mode
#232338# WLAN MAC Address
#232339# WLAN Test Mode
#0842# Vibra Motor Test Mode
#0782# Real Time Clock Test
#0673# Audio Test Mode
#0# General Test Mode
#2263# RF Band Selection
#872564# USB Logging Control
#4238378# GCF Configuration
#0283# Audio Loopback Control
#1575# GPS Control Menu
#3214789650# LBS Test Mode
#44336# Sofware Version Info
#7780# Factory Reset
27673855# Full Factory Reset
#0289# Melody Test Mode
#2663# TSP / TSK firmware update
#03# NAND Flash S/N
#0589# Light Sensor Test Mode
#0588# Proximity Sensor Test
#3282727336# Data Usage Status
#7594# Remap Shutdown to End Call TSK
#34971539# Camera Firmware
#528# WLAN Engineering Mode
#7412365# Camera Firmware Menu
#07# Test History
#3214789# GCF Mode Status
#272886# Auto Answer Selection
#8736364# OTA Update Menu
#301279# HSDPA/HSUPA Control Menu
#7353# Quick Test Menu
27674387264636# Sellout SMS / PCODE view
#7465625# View Phone Lock Status
7465625638# Configure Network Lock MCC/MNC
#7465625638# Insert Network Lock Keycode
##7780##* Factory data reset - Clears Google-account data, system and program settings and installed programs. system will not be deleted, and OEM programs, as well as My Documents (pictures, music, videos)
Does anyone know any other neat tricks like this?