Nobody is laughing at you. By "for laughs" (a colloquialism in the English language) I simply meant "just do this extra thing for us because it might (or might not) prove to be helpful."
Somebody will jump in here any minute to give you detailed instructions for capturing a screen shot. You will make life easier for that person if you tell us which operating system and version you are using.
If necessary, crop the captured image in Photoshop (with no re-sampling) to show only the pertinent area(s), then use Save for Web to create a PNG for upload here.
OK, now we're getting somewhere.
Can you select, say, just the orange section at upper left -- by either ungrouping first or using the direct selection tool -- and show us the appearance panel under that condition?
That blue thingy, I believe, is an artifact of the capture. I can't recall its significance at the moment. You can just ignore it.
EDIT: Upon closer inspection, the blue thingy seems to be an object (with no fill or stroke) in your Illustrator artwork. I have no idea how it got there. You should be able to direct-select it and delete it.
What happens if you select everything and click the Reduce to Basic Appearance button (I cannot see it in the screenshot, and the panel looks different from what I see in 10, so I have no clue what it should look like, but the buttons should declare themselves upon hovering)?
Maybe some of the parts have shadows applied, others not.
You may save yourself with an Undo (Ctrl/Cmd+Z).
Or you could roll the whole artwork so it does not protrude.
From the appearance panel capture with all selected (post #15), it is possible the OP has some grouped paths, in which case Reduce to Basic Appearance will not be available as an option. In AI 14 (CS4), I believe 'Reduce to Basic Appearance' can only be selected via the appearance panel's flyout menu (i.e., there is no dedicated button). It will be grayed out if there are any grouped objects.
First try this. Select the small circular art on the left side and delete it. If you cannot use the lasso tool to just select that then use the layers panel to select the art and the delete it.
Once deleted select your art and see if it reads in the Appearance panel as a group and if the groups has the effect applied to it.
if so delete the drop shadow effect.
If not open the layers and elect the layer the art is one go back the appearance panle to see if it now lists the layer as the top item and see if the effect is applied to the layer if so select it and then delete it.
if none of that works then use the direct select tool and select each elemental path, one at a time and delete the drop shadow effect using the appearance panel.
One of these methods will work. Unless the drop shadow is actually a filter and not an effect. In that event use the direct select tool to select the select the shadow on the canvas and then delete it.
I've just gone to do the task you requested and for some reason I cannot ungroup the object BUT 'shadow' is now in the Appearance panel! How has this happened??! I click on it and taaa daaaa - the shadow is gone! Woop woop! Another problem - because I cannot ungroup them I can't get rid of the 'blue thing'.
Thank you for all your help Harron, you've been a star.
Another problem - because I cannot ungroup them I can't get rid of the 'blue thing'.
You cannot select it with the Direct Selection Tool (by dragging over it or clicking the individual Anchor Points) and delete it?