what malware scans has she run? I would update and run malwarebytes, and do this online scan to rule out any type of malware causing the problem.
Malwarebytes' Anti-Malware free
FREE Online Virus Scan
and another good choice is
SUPERAntiSpyware Free Edition

please let us know the results of those.

also has she used the secunia tool to make sure everything is fully updated?
Secunia Online Software Inspector

on the occasion when she can get it to operate have her go to start, control panel, administrative tools, event viewer and look at each item listed there double click and see if there are any red or yellow flags, if so double click on those entries for a dialog box with details the error code numbers can be googled to find out more info on what is causing the events.
How To Use the Event Viewer Applet

I neglected to mentioned that we have been working on the problem for a couple of weeks and she ran Superantispyware, Malwarebytes and did a Trend housecall before formatting and reinstalling the operating system the first time. She does use Secunia and everything was up to date . I do not remember asking her to check the event viewer so will have her do that when I talk to her tonight.

Bob try Googling "windows opens with no icons or taskbar" without the quotes. It brings up some interesting sites. I would not suggest any one site particularly but there seemed some valuable information there. They mostly seem to involve adapting the registry so just make sure you create a back up before making any changes using regedit

Thanks Owbist. We had googled the problem many times but I hadn't noticed suggestions regarding the registry so I added the word registry in the advanced search and found some interesting ones. Many of them involve getting rid of a virus or other malware. I can't believe that those could survive two formats and re-installs of the OS. I don't mind walking her through editing the registry if I find a suggestion that seems to fit. Intermittent problems are a pain. This one happens most of the time but not every time. I wish I could get up there and get my hands on the computer but even then I don't know what I would do.

Bob using the quote I mentioned - windows opens with no icons or taskbar - brought up over 130,000 pages. I would discount the virus theory seeing you have wiped the drive.

You last post suggests distance is causing a problem. Try Teamviewer, it is brilliant. You click to 'Start full version' and you daughter clicks 'Join a session'. She will then give you 3 sets of 3 numbers to punch in, you then click 'Connect to partner' and she gives you a 4 digit password. Once connected you have her screen on your computer and total control of her mouse and keyboard movements. You can also use your own computer if you need to search for stuff while working on the other confuser.

She can also use her mouse and keyboard if needed. You can also reboot her computer and re-connect automatically. You cannot re-connect once both of you have broken the connection, this is her security. I use Skype to talk with the person whose computer I am working on so they are kept up to date on where and what I am doing. Any VOIP program will do that of course.

Here is a link that might be useful: Teamviewer

Thanks Owbist. That will be our next step. Right now we found that in event viewer there are several disk event 11 errors in the system category. This seems to indicate a problem with the disk controller, cable or disk. Since the Dell diagnostics didn't find a problem with the disk the next step will be to try a new cable which would be the easiest solution if it works.

does her pc have a card reader on it? if so that may be what is causing those errors I have heard of that being the cause several times, if it does have a card reader try going to device manager and remove it from there see if that helps, it may need to be disconnected. It can also happen when a stand alone usb card reader is connected to the pc with out a card in any of the slots.

and of course it could be driver related.
Please let us know your outcomes

Googling found a suggestion that may have worked. The computer booted up successfully 3 times in a row which hasn't happened for some time. We went to the device manager and changed the ATA/ATAPI controller to use PIO instead of DMA. Hope it continues to work.

Hmm that is odd. Usually one makes the change the other way. Glad it worked Bob, we'll keep our fingers crossed that it stays fixed now.