You can open the recent apps list using adb.
Setup ADB in PC, connect the phone into PC, enable USB debugging in phone, launch a shell on PC, and enter:
adb shell input keyevent --longpress 3
3 is a KeyEvent to call Home. See more of them here. Also, if the recent list isn't bind to long press Home key but to another key like Menu/Overview key then the command wouldn't result in any visible activity on device. In that case, look for the KeyEvent associated to your key.
--longpress isn't available to Android versions prior to Kitkat release so the command would give you error on ICS or JB.
Alternatively, in my devices I found that the activity responsible for bringing that list is managed by SystemUI. You can call that activity using ADB to show that recent apps list.
adb shell am start -n com.android.systemui/.recent.RecentsActivity
--activity-clear-task at the end if the activity doesn't show up in front. Space is important before
This worked on Android 4.2.1, 4.4.2 and 5.0.2. For Lollipop, use
com.android.systemui/.recents.RecentsActivity to show up recent apps list in 3D, otherwise, you would see a 2D list which is usual for Android 4.x.
As for launching it within the device without entering the commands, simply use QuickShortcutMaker or Tasker to make home screen shortcuts for the said activity.