Well, of course. Imagine trying to run any business without taking inventory of your products. You'd go broke, literally, before you knew it. I recently asked a manager for a list of the applications supported by his team, and was expecting a list of about a dozen. It took several weeks to get me the list because they didn't have it and couldn't recite the applications off from memory.
An asset inventory does not have to be a complicated thing; it could be a simple spreadsheet with one column listing the names of the products you support. In the adjacent columns you can include any related data you want, but I'd make a few suggestions. Have columns that display the base technologies and architecture. Are the applications Java, .Net, C++, unknown (vendor supplied). Are the applications monolithic, batch, client/server, web, or mobile?
Include some columns to indicate who is responsible with phone and pager numbers. Have a column or two to indicate the number of outages this year, bugs, or Help Desk tickets that have been opened. Make sure you know when the last update was made and when the next one is scheduled.
At first, this data will seem obvious and of little value. Over time, as you add columns, you'll find great value. For instance, if you have two columns for each year or quarter and you track the number of changes made (you decide how to count that one) and the number of problems reported, you'll have a pretty useful quality metric that you can share with your business users (or not...!).
Start simple, with just a list of the applications and add additional columns/data over time. Don't complicate things, but record the basics.